Nexcelsior

Wednesday, 13th Dec, 2023 by Krilling for Company in Integer Sequences, Mathematics, Triangular Numbers and tagged A006003, excelsior, OEIS, Online Encyclopedia of Integer Sequences, sum of next n integers | Leave a comment

In “The Trivial Troot”, I looked at what happens when tri(k), the k-th triangular number, is one digit longer than the previous triangular number, tri(k-1):

6 = tri(3) 10 = tri(4)
91 = tri(13) 105 = tri(14) 990 = tri(44) 1035 = tri(45) [...] ↓
10 ← 4 105 ← 14 1035 ← 45 10011 ← 141 100128 ← 447 1000405 ← 1414 10001628 ← 4472 100005153 ← 14142 1000006281 ← 44721 10000020331 ← 141421 100000404505 ← 447214 1000001326005 ← 1414214 10000002437316 ← 4472136 100000012392316 ← 14142136 [...]

What’s going on with k? In a sense, it’s calculating the square roots of 2 and 20:

√2 = 1·414213562373095048801688724209698078569671875376948073176679738... √20 = 4·472135954999579392818347337462552470881236719223051448541794491...

Now let’s say “Excelsior!” and go higher with a related sequence. A006003 is defined at the Online Encyclopedia of Integer Sequences as the “sum of the next n natural numbers”. Here it is:

1 = 1 5 = 2 + 3 15 = 4 + 5 + 6 34 = 7 + 8 + 9 + 10 65 = 11 + 12 + 13 + 14 + 15 111 = 16 + 17 + 18 + 19 + 20 + 21 175 = 22 + 23 + 24 + 25 + 26 + 27 + 28 260 = 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 369 = 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 505 = 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 671 = 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 870 = 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 1105 = 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 + 90 + 91 1379 = 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 + 104 + 105 1695 = 106 + 107 + 108 + 109 + 110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 + 118 + 119 + 120 2056 = 121 + 122 + 123 + 124 + 125 + 126 + 127 + 128 + 129 + 130 + 131 + 132 + 133 + 134 + 135 + 136 2465 = 137 + 138 + 139 + 140 + 141 + 142 + 143 + 144 + 145 + 146 + 147 + 148 + 149 + 150 + 151 + 152 + 153 2925 = 154 + 155 + 156 + 157 + 158 + 159 + 160 + 161 + 162 + 163 + 164 + 165 + 166 + 167 + 168 + 169 + 170 + 171 3439 = 172 + 173 + 174 + 175 + 176 + 177 + 178 + 179 + 180 + 181 + 182 + 183 + 184 + 185 + 186 + 187 + 188 + 189 + 190 4010 = 191 + 192 + 193 + 194 + 195 + 196 + 197 + 198 + 199 + 200 + 201 + 202 + 203 + 204 + 205 + 206 + 207 + 208 + 209 + 210 [...]

If you’re familiar with triangular numbers, you’ll see that sumnext(k) is always higher than tri(k), except for sumnext(1) = 1 = tri(k). Now, this is what happens when sumnext(k) is one digit longer than sumnext(k-1):

5 = sumnext(2) 15 = sumnext(3)
65 = sumnext(5) 111 = sumnext(6) 870 ← 12 1105 ← 13 9855 ← 27 10990 ← 28 97585 ← 58 102719 ← 59 976625 ← 125 1000251 ← 126 9951391 ← 271 10061960 ← 272 99588644 ← 584 100101105 ← 585 997809119 ← 1259 1000188630 ← 1260 9995386529 ← 2714 10006439295 ← 2715 [...] ↓
15 ← 3 111 ← 6 1105 ← 13 10990 ← 28 102719 ← 59 1000251 ← 126 10061960 ← 272 100101105 ← 585 1000188630 ← 1260 10006439295 ← 2715 100049490449 ← 5849 1000188006300 ← 12600 10000910550385 ← 27145 100003310078561 ← 58481 1000021311323825 ← 125993 10000026341777165 ← 271442 100000232056567634 ← 584804 1000002262299152685 ← 1259922 10000004237431278525 ← 2714418 100000026858987459346 ← 5848036 1000000119305407615071 ← 12599211 10000000921801015908705 ← 27144177 100000001209342964609615 ← 58480355 1000000000250317736274865 ← 125992105 10000000037633414521952245 ← 271441762 100000000183357362892853070 ← 584803548 1000000000250317673908773025 ← 1259921050 [...]

What’s going on now? In a sense, the digits of k are approximating the cube roots of 20, 200 and 2000:

2.714417616594906571518089469679489204805107769489096957284365443... = cuberoot(20) 5.848035476425732131013574720275845557060997270202060082845147020... = cuberoot(200) 12.59921049894873164767210607278228350570251464701507980081975112... = cuberoot(2000)
cuberoot(20) = 2.714417616594906571518089469679489204805107769489096957284365443... cuberoot(200) = 5.848035476425732131013574720275845557060997270202060082845147020... cuberoot(2000) = 12.59921049894873164767210607278228350570251464701507980081975112...

So you could say that this sequence has gone nexcelsior: sumnext(k) > tri(k); cubes are higher than squares; and (20, 200, 2000) is bigger than (2, 20).

Previously Pre-Posted…

• “The Trivial Troot” — explaining the earlier pattern in triangular numbers

Pirouetting the Perimeter

Monday, 11th Dec, 2023 by Krilling for Company in Geometry, Mathematics, Triangles and tagged animated gifs, equilateral triangle, perimeters, pirouettes, point moving on perimeter, recreational math, recreational mathematics, recreational maths | Leave a comment

Imagine a ballerina pirouetting on the perimeter of an equilateral triangle. Suppose that her armlength is half the radius of the circumscribed circle. If her right arm is represented in green and her right hand in yellow, this is one path that her right hand might trace, depending on the relative speeds of her feet and her pirouettes:

The perimeter of an equilateral triangle

A ballerina pirouetting on the perimeter stage #1

Perimeter pirouette #2

Perimeter pirouette #3

Perimeter pirouette #4

Perimeter pirouette #5

Perimeter pirouette #6

Perimeter pirouette #7

Perimeter pirouette #8

Perimeter pirouette #9

Perimeter pirouette #10

[…]

↓

[…]

Perimeter pirouette #128

Perimeter pirouettes (animated)

The right hand of the ballerina during the pirouettes

The full path traced by the ballerina’s right hand

The paths change as you adjust speed and direction (clockwise or anticlockwise) of the pirouettes, the armlength of the ballerina, and so on:

speed = 0.5 (pirouettes)

speed = 0.5 (path)

speed = -1 (anticlockwise pirouettes)

speed = -1 (path)

speed = 1, armlength = 1/3 (pirouettes)

speed = 1, armlength = 1/3 (path)

speed = -0.5 (pirouettes)

speed = -0.5 (path)

speed = -0.75 (pirouettes)

speed = -0.75 (path)

And what about the paths produced by pirouettes on the perimeters of other polygons? Watch this space.

Grow Fourth

Friday, 8th Dec, 2023 by Krilling for Company in Arithmetic, Integer Sequences, Mathematics, Powers and tagged 81, Arithmetic, David Wells, fourth powers, groups of integers, integer sums, Penguin Dictionary of Curious and Interesting Numbers, powers, recreational math, recreational mathematics, recreational maths | Leave a comment

Write the integers in groups of one, two, three, four… numbers like this:

1, 2,3, 4,5,6, 7,8,9,10, 11,12,13,14,15, 16,17,18,19,20,21, 22,23,24,25,26,27,28, 29,30,31,32,33,34,35,36, 37,38,39,40,41,42,43,44,45, 46,47,48,49,50,51,52,53,54,55, 56,57,58,59,60,61,62,63,64,65,66…

Now delete every second group:

1, ~~2,3,~~ 4,5,6, ~~7,8,9,10,~~ 11,12,13,14,15, ~~16,17,18,19,20,21,~~ 22,23,24,25,26,27,28, ~~29,30,31,32,33,34,35,36,~~ 37,38,39,40,41,42,43,44,45, ~~46,47,48,49,50,51,52,53,54,55~~, 56,57,58,59,60,61,62,63,64,65,66…

↓↓↓

1, 4,5,6, 11,12,13,14,15, 22,23,24,25,26,27,28, 37,38,39,40,41,42,43,44,45, 56,57,58,59,60,61,62,63,64,65,66…

The sum of the first n remaining groups equals n^4:

1 = 1 = 1^4

1 + 4+5+6 = 16 = 2^4

1 + 4+5+6 + 11+12+13+14+15 = 81 = 3^4

1 + 4+5+6 + 11+12+13+14+15 + 22+23+24+25+26+27+28 = 256 = 4^4

1 + 4+5+6 + 11+12+13+14+15 + 22+23+24+25+26+27+28 + 37+38+39+40+41+42+43+44+45 = 625 = 5^4

1 + 4+5+6 + 11+12+13+14+15 + 22+23+24+25+26+27+28 + 37+38+39+40+41+42+43+44+45 + 56+57+58+59+60+61+62+63+64+65+66 = 1296 = 6^4

1 + 4+5+6 + 11+12+13+14+15 + 22+23+24+25+26+27+28 + 37+38+39+40+41+42+43+44+45 + 56+57+58+59+60+61+62+63+64+65+66 + 79+80+81+82+83+84+85+86+87+88+89+90+91 = 2401 = 7^4

1 + 4+5+6 + 11+12+13+14+15 + 22+23+24+25+26+27+28 + 37+38+39+40+41+42+43+44+45 + 56+57+58+59+60+61+62+63+64+65+66 + 79+80+81+82+83+84+85+86+87+88+89+90+91 + 106+107+108+109+110+111+112+113+114+115+116+117+118+119+120 = 4096 = 8^4

From David Wells’ Penguin Dictionary of Curious and Interesting Numbers (1986), entry for “81”

Back to LIFE

Monday, 20th Nov, 2023 by Krilling for Company in Arithmetic, Geometry, Mathematics, Squares and tagged animated gifs, game of life, grid of squares, John Conway, John Horton Conway, life, mathematical game, mathematical games, recreational math, recreational mathematics, recreational maths, square spiral | Leave a comment

As pre-previously described on OotÜ-F, the English mathematician John Conway invented the Game of Life. It’s played on a grid of squares with counters. First you put counters on the grid in any pattern you please, random or regular, then you add or remove counters according to three simple rules applied to each square of the grid:

1. If an empty square has exactly three counters as neighbors, put a new counter on the square.
2. If a counter has two or three neighbors, leave it where it is.
3. If a counter has less than two or more than three neighbors, remove it from the grid.

There are lots of variants on Life and I wondered what would happen if you turned the grid into a kind of two-dimensional Pascal’s triangle. You start with 1 in the central square, then apply this rule to each square, [x,y], of the grid:

1. Add all numbers in the eight squares surrounding [x,y], then put that value in [x,y] (as soon as you’ve summed all other squares).

When a square is on the edge of the grid, its [x] or [y] value wraps to the opposite edge. Here’s this Pascal’s Life in action:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Pascal's square #1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Pascal's square #2 0 0 0 0 0 0 0 0 1 2 3 2 1 0 0 2 2 4 2 2 0 0 3 4 8 4 3 0 0 2 2 4 2 2 0 0 1 2 3 2 1 0 0 0 0 0 0 0 0 Pascal's square #3 01 03 06 07 06 03 01 03 06 12 12 12 06 03 06 12 27 27 27 12 06 07 12 27 24 27 12 07 06 12 27 27 27 12 06 03 06 12 12 12 06 03 01 03 06 07 06 03 01 Pascal's square #4 021 038 056 067 056 038 021 038 070 100 124 100 070 038 056 100 132 168 132 100 056 067 124 168 216 168 124 067 056 100 132 168 132 100 056 038 070 100 124 100 070 038 021 038 056 067 056 038 021 Pascal's square #5 0285 0400 0560 0615 0560 0400 0285 0400 0541 0755 0811 0755 0541 0400 0560 0755 1070 1140 1070 0755 0560 0615 0811 1140 1200 1140 0811 0615 0560 0755 1070 1140 1070 0755 0560 0400 0541 0755 0811 0755 0541 0400 0285 0400 0560 0615 0560 0400 0285 Pascal's square #6 2996 3786 4697 5176 4697 3786 2996 3786 4785 5892 6525 5892 4785 3786 4697 5892 7153 7941 7153 5892 4697 5176 6525 7941 8840 7941 6525 5176 4697 5892 7153 7941 7153 5892 4697 3786 4785 5892 6525 5892 4785 3786 2996 3786 4697 5176 4697 3786 2996 Pascal's square #7

As you can see, the numbers quickly get big, so I adjusted the rule: sum the eight neighbors of [x,y], then put sum modulo 10 in [x,y]. The modulus of a number, n is its remainder when it’s divided by another number. For example, 3 modulo 10 = 3, 7 modulo 10 = 7, 10 modulo 10 = 0, 24 modulo 10 = 4, and so on. Pascal’s Life modulo 10 looks like this:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Pascal's square (n mod 10) #1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Pascal's square (n mod 10) #2 0 0 0 0 0 0 0 0 1 2 3 2 1 0 0 2 2 4 2 2 0 0 3 4 8 4 3 0 0 2 2 4 2 2 0 0 1 2 3 2 1 0 0 0 0 0 0 0 0 Pascal's square (n mod 10) #3 1 3 6 7 6 3 1 3 6 2 2 2 6 3 6 2 7 7 7 2 6 7 2 7 4 7 2 7 6 2 7 7 7 2 6 3 6 2 2 2 6 3 1 3 6 7 6 3 1 Pascal's square (n mod 10) #4 1 8 6 7 6 8 1 8 0 0 4 0 0 8 6 0 2 8 2 0 6 7 4 8 6 8 4 7 6 0 2 8 2 0 6 8 0 0 4 0 0 8 1 8 6 7 6 8 1 Pascal's square (n mod 10) #5 5 0 0 5 0 0 5 0 1 5 1 5 1 0 0 5 0 0 0 5 0 5 1 0 0 0 1 5 0 5 0 0 0 5 0 0 1 5 1 5 1 0 5 0 0 5 0 0 5 Pascal's square (n mod 10) #6 6 6 7 6 7 6 6 6 5 2 5 2 5 6 7 2 3 1 3 2 7 6 5 1 0 1 5 6 7 2 3 1 3 2 7 6 5 2 5 2 5 6 6 6 7 6 7 6 6 Pascal's square (n mod 10) #7 7 5 3 3 3 5 7 5 9 5 1 5 9 5 3 5 1 7 1 5 3 3 1 7 6 7 1 3 3 5 1 7 1 5 3 5 9 5 1 5 9 5 7 5 3 3 3 5 7 Pascal's square (n mod 10) #8

Now add graphics and use n modulo 2 (where all even numbers → 0 and all odd numbers → 1). If you start with a 17×17 square with a square pattern of 1s, you’ll see it evolve like this when 0s are represented in black and 1s are represented in red:

n mod 2 on 17×17 square #1

n mod 2 #2

n mod 2 #3

n mod 2 #4

n mod 2 #5

n mod 2 #6

n mod 2 #7

n mod 2 #8

n mod 2 #9

n mod 2 #10

n mod 2 #11

n mod 2 #12

n mod 2 #13

n mod 2 #14

n mod 2 #15

n mod 2 #16

n mod 2 (animated)

As you can see, the original square re-appears. So do other patterns. Here’s an animated gif for n modulo 2 seeded with a pattern of 1s spelling LIFE:

Now try a spiral as the seed:

Spiral with n mod 2 #1

Spiral with n mod 2 #2

Spiral with n mod 2 #3

Spiral with n mod 2 #4

Spiral with n mod 2 #5

Spiral with n mod 2 #6

Spiral with n mod 2 #7

Spiral with n mod 2 #8

Spiral with n mod 2 #9

Spiral with n mod 2 #10

Spiral with n mod 2 #11

Spiral mod 2 (animated)

Now try the same pattern using modulo 3, where 0s are represented in black, 1s are represented in red and 2s in green. The pattern returns with different colors, i.e. with different underlying digits:

Spiral mod 3 on 27×27 square #1

Spiral mod 3 #2

Spiral mod 3 #3

Spiral mod 3 #4

Spiral mod 3 #5

Spiral mod 3 #6

Spiral mod 3 #7

Spiral mod 3 #8

Spiral mod 3 #9

Spiral mod 3 #10

Spiral mod 3 #11

⇓

[…]

Spiral mod 3 #19

⇓

[…]

Spiral mod 3 #28

⇓

[…]

Spiral mod 3 #37

⇓

[…]

Spiral mod 3 #46

Spiral mod 3 (animated)

LIFE mod 3 (animated)

Now try n modulo 5, with 0s represented in black, 1s represented in red, 2s in green, 3s in yellow and 4s in dark blue. Again the pattern returns in different colors:

Spiral mod 5 on 25×25 square #1

Spiral mod 5 #2

Spiral mod 5 #3

Spiral mod 5 #4

Spiral mod 5 #5

Spiral mod 5 #6

⇓

[…]

Spiral mod 5 #26

⇓

[…]

Spiral mod 5 #31

⇓

[…]

Spiral mod 5 #76

⇓

[…]

Spiral mod 5 #81

Spiral mod 5 (animated)

Finally, try a svastika modulo 7, with 0s represented in black, 1s represented in red, 2s in green, 3s in yellow, 4s in dark blue, 5s in purple and 6s in light blue:

Svastika mod 7 on 49×49 square #1

Svastika mod 7 #2

Svastika mod 7 #3

Svastika mod 7 #4

Svastika mod 7 #5

Svastika mod 7 #6

Svastika mod 7 #7

Svastika mod 7 #8

⇓

[…]

Svastika mod 7 #15

⇓

[…]

Svastika mod 7 #22

⇓

[…]

Svastika mod 7 #29

⇓

[…]

Svastika mod 7 #36

⇓

[…]

Svastika mod 7 #43

Svastika mod 7 (animated)

Previously Pre-Posted…

• Eternal LIFE — a first look at the Game of Life

Eternal LIFE

Thursday, 9th Nov, 2023 by Krilling for Company in Geometry, Infinity, Mathematics, Programming and tagged animated gifs, blinker, cellular automata, eternal recurrence, French language, Friedrich Nietzsche, game of life, geometry, grid of squares, John Conway, John Horton Conway, life, mathematical game, quotes about mathematics, recreational math, recreational mathematics, recreational maths, Siméon-Denis Poisson, square spiral | 2 Comments

The French mathematician Siméon-Denis Poisson (1781-1840) once said: « La vie n’est bonne qu’à deux choses : à faire des mathématiques et à les professer. » — “Life is good only for two things: doing mathematics and teaching mathematics.” The German philosopher Nietzsche wouldn’t have agreed. He thought (inter alia) that we must learn to accept life as eternally recurring. Everything we do and experience will happen again and again for ever. Can you accept life like that? Then your life is good.

But neither Poisson or Nietzsche knew that Life, with a capital L, would take on a new meaning in the 20th century. It became a mathematical game played on a grid of squares with counters. You start by placing counters in some pattern, regular or random, on the grid, then you add or remove counters according to three simple rules applied to each square of the grid:

1. If an empty square has exactly three counters as neighbors, put a new counter on the square.
2. If a counter has two or three neighbors, leave it where it is.
3. If a counter has less than two or more than three neighbors, remove it from the grid.

And there is a meta-rule: apply all three rules simultaneously. That is, you check all the squares on the grid before you add or remove counters. With these three simple rules, patterns of great complexity and subtlety emerge, growing and dying in a way that reminded the inventor of the game, the English mathematician John Conway, of living organisms. That’s why he called the game Life.

Let’s look at Life in action, with the seeding counters shown in green. Sometimes the seed will evolve and disappear, sometimes it will evolve into one or more fixed shapes, sometimes it will evolve into dynamic shapes that repeat again and again. Here’s an example of a seed that evolves and disappears:

Seeded with cross (arms 4+1+4) stage #1

Life stage #2

Life stage #3

Life stage #4

Life stage #5

Life stage #6

Life stage #7

Death at stage #8

Life from cross (animated)

The final stage represents death. Now here’s a cross that evolves towards dynamism:

Life seeded with cross (arms 3+1+3) stage #1

Life stage #2

Life stage #3

Life stage #4

Life stage #5

Life stage #6 (same as stage #4)

Life stage #7 (same as stage #5)

Life stage #8 (same as stage #4 again)

Life from cross (animated)

A line of three blocks swinging between horizontal and vertical is called a blinker:

Four blinkers

And here’s a larger cross that evolves towards stasis:

Life seeded with cross (arms 7+1+7) stage #1

Life stage #2

Life stage #3

Life stage #4

Life stage #5

Life stage #6

Life stage #7

Life stage #8

Life stage #9

Life stage #10

Life stage #11

Life stage #12

Life stage #13

Life stage #14

Life stage #15

Life stage #16

Life from cross (animated)

This diamond with sides of 24 blocks evolves towards even more dynamism:

Life from 24-sided diamond (animated)

Looping Life from 24-sided diamond (animated)

The game of Life obviously has many variants. In the standard form, you’re checking all eight squares around the square whose fate is in question. If that square is (x,y), these are the eight other squares you check:

(x+1,y+1), (x+0,y+1), (x-1,y+1), (x-1,y+0), (x-1,y-1), (x+0,y-1), (x+1,y-1), (x+1,y+0)

Now trying checking only four squares around (x,y), the ones above and below and to the left and the right:

(x+1,y+1), (x-1,y+1), (x-1,y-1), (x+1,y-1)

And apply a different set of rules:

1. If a square has one or three neighbors, it stays alive or (if empty) comes to life
2. Otherwise the square remains or becomes empty.

With that check and those rules, the seed first disappears, then re-appears, for ever (note that the game is being played on a torus):

Evolution of spiral seed

Eternally recurring spiral

This happens with any seed, so you can use Life to bring Nietzsche’s eternal recurrence to life:

Evolution of LIFE

Eternally recurring LIFE

Pyramidic Palindromes

Tuesday, 17th Oct, 2023 by Krilling for Company in Base Behavior, Integer Sequences, π, Mathematics, Palindromic numbers, Pi, Powers, Squares and tagged base 10, base 9, cannonballs, cubic pyramidal numbers, East Sussex, palindromes, pi, pile of cannonballs, Rye Castle, square pyramidal numbers, sum of cubes, sum of squares | Leave a comment

As I’ve said before on Overlord of the Über-Feral: squares are boring. As I’ve shown before on Overlord of the Über-Feral: squares are not so boring after all.

Take A000330 at the Online Encyclopedia of Integer Sequences:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, 16206, 17575, 19019, 20540, 22140, 23821, 25585, 27434, 29370… — A000330 at OEIS

The sequence shows the square pyramidal numbers, formed by summing the squares of integers:

• 1 = 1^2
• 5 = 1^2 + 2^2 = 1 + 4
• 14 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9
• 30 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16

[…]

You can see the pyramidality of the square pyramidals when you pile up oranges or cannonballs:

Square pyramid of 91 cannonballs at Rye Castle, East Sussex (Wikipedia)

I looked for palindromes in the square pyramidals. These are the only ones I could find:

1 (k=1)
5 (k=2)
55 (k=5)
1992991 (k=181)

The only ones in base 10, that is. When I looked in base 9 = 3^2, I got a burst of pyramidic palindromes like this:

1 (k=1)
5 (k=2)
33 (k=4) = 30 in base 10 (k=4)
111 (k=6) = 91 in b10 (k=6)
122221 (k=66) = 73810 in b10 (k=60)
123333321 (k=666) = 54406261 in b10 (k=546)
123444444321 (k=6,666) = 39710600020 in b10 (k=4920)
123455555554321 (k=66,666) = 28952950120831 in b10 (k=44286)
123456666666654321 (k=666,666) = 21107018371978630 in b10 (k=398580)
123456777777777654321 (k=6,666,666) = 15387042129569911801 in b10 (k=3587226)
123456788888888887654321 (k=66,666,666) = 11217155797104231969640 in b10 (k=32285040)

The palindromic pattern from 6[…]6 ends with 66,666,666, because 8 is the highest digit in base 9. When you look at the 666,666,666th square pyramidal in base 9, you’ll find it’s not a perfect palindrome:

123456801111111111087654321 (k=666,666,666) = 8177306744945450299267171 in b10 (k=290565366)

But the pattern of pyramidic palindromes is good while it lasts. I can’t find any other base yielding a pattern like that. And base 9 yields another burst of pyramidic palindromes in a related sequence, A000537 at the OEIS:

1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176, 90000, 105625, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 314721, 354025, 396900, 443556, 494209, 549081… — A000537 at OEIS

The sequence is what you might call the cubic pyramidal numbers, that is, the sum of the cubes of integers:

• 1 = 1^2
• 9 = 1^2 + 2^3 = 1 + 8
• 36 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27
• 100 = 1^3 + 2^3 + 3^3 + 4^3 = 1 + 8 + 27 + 64

[…]

I looked for palindromes there in base 9:

1 (k=1) = 1 (k=1)
121 (k=4) = 100 in base 10 (k=4)
12321 (k=14) = 8281 (k=13)
1234321 (k=44) = 672400 (k=40)
123454321 (k=144) = 54479161 (k=121)
12345654321 (k=444) = 4412944900 (k=364)
1234567654321 (k=1444) = 357449732641 (k=1093)
123456787654321 (k=4444) = 28953439105600 (k=3280)
102012022050220210201 (k=137227) = 12460125198224404009 (k=84022)

But while palindromes are fun, they’re not usually mathematically significant. However, this result using the square pyrmidals is certainly significant:

Previously Pre-Posted…

More posts about how squares aren’t so boring after all:

• Curvous Energy
• Back to Drac #1
• Back to Drac #2
• Square’s Flair

Blancmange Butterfly

Friday, 29th Sep, 2023 by Krilling for Company in Blancmange Curves, Circles, Fractals, Geometry, Mathematics and tagged animated gifs, animated gifs of fractals, arching blancmange circle, arching blancmange curve, arcs, blancmange circles, blancmange fractal, Circles, curves, fractal butterfly, fractal geometry, fractals, geometry, recreational math, recreational mathematics, recreational maths | Leave a comment

Blancmange butterfly. Is that a ’60s psychedelic band? No, it’s one of the shapes you can get by playing with blancmange curves. As I described in “White Rites”, a blancmange curve is a fractal created by summing the heights of successively smaller and more numerous zigzags, like this:

Zigzags 1 to 10

Zigzags 1 to 10 (animated)

Blancmange curve

In the blancmange curves below, the height (i.e., the y co-ordinate) has been normalized so that all the images are the same height:

Construction of a normalized blancmange curve (animated)

This is the solid version:

Solid normalized blancmange curve (animated)

I wondered what happens when you wrap a blancmange curve around a circle. Well, this happens:

Construction of a blancmange circle (animated)

You get what might be called a blancmange butterfly. The solid version looks like this (patterns in the circles are artefacts of the graphics program I used):

Solid blancmange circle (animated)

Next I tried using arcs rather zigzags to construct the blancmange curves and blancmange circles:

Arching blancmange curve (i.e., constructed with arcs) (animated)

And below is the circular version of a blancmange curve constructed with arcs. The arching circular blancmanges look even more like buttocks and then intestinal villi (the fingerlike projections lining our intestines):

Arching blancmange circle (animated)

The variations on blancmange curves don’t stop there — in fact, they’re infinite. Below is a negative arching blancmange curve, where the heights of the original arching blancmange curve are subtracted from the (normalized) maximum height:

Negative arching blancmange curve (animated)

And here’s an arching blancmange curve that’s alternately negative and positive:

Negative-positive arching blancmange curve (animated)

The circular version looks like this:

Negative-positive arching blancmange circle (animated)

Finally, here’s an arching blancmange curve that’s alternately positive and negative:

Positive-negative arching blancmange curve (animated)

And the circular version:

Positive-negative arching blancmange circle (animated)

Elsewhere Other-Accessible…

• White Rites — more variations on blancmange curves

Nuts for Numbers

Friday, 22nd Sep, 2023 by Krilling for Company in Arithmetic, Base Behavior, Integer Sequences, Mathematics and tagged 210, 2772, Arithmetic, base 10, base 8, base 9, integer sums, Mr Logic, Mr Logic and nuts, nuts, nutty numbers, nutty sums, palindromes, recreational math, recreational mathematics, recreational maths, summing integers, sums of consecutive integers, Viz, Viz comic | Leave a comment

I was looking at palindromes created by sums of consecutive integers. And I came across this beautiful result:

2772 = sum(22..77)

2772 = 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77

You could call 2772 a nutty sum, because 77 is held inside 22 like a kernel inside a nutshell. Here some more nutty sums, sum(n₁..n₂), where n₂ is a kernel in the shell of n₁:

1599 = sum(19..59)
2772 = sum(22..77)
22113 = sum(23..211)
159999 = sum(199..599)
277103 = sum(203..771)
277722 = sum(222..777)
267786 = sum(266..778)
279777 = sum(277..797)
1152217 = sum(117..1522)
1152549 = sum(149..1525)
1152767 = sum(167..1527)
4296336 = sum(436..2963)
5330303 = sum(503..3303)
6235866 = sum(626..3586)
8418316 = sum(816..4183)
10470075 = sum(1075..4700)
11492217 = sum(1117..4922)
13052736 = sum(1306..5273)
13538277 = sum(1377..5382)
14557920 = sum(1420..5579)
15999999 = sum(1999..5999)
25175286 = sum(2516..7528)
26777425 = sum(2625..7774)
27777222 = sum(2222..7777)
37949065 = sum(3765..9490)
53103195 = sum(535..10319)
111497301 = sum(1101..14973)

Of course, you can go the other way and find nutty sums where sum(n₁..n₂) produces n₁ as a kernel inside the shell of n₂:

147 = sum(4..17)
210 = sum(1..20)
12056 = sum(20..156)
13467 = sum(34..167)
22797 = sum(79..227)
22849 = sum(84..229)
26136 = sum(61..236)
1145520 = sum(145..1520)
1208568 = sum(208..1568)
1334667 = sum(334..1667)
1540836 = sum(540..1836)
1931590 = sum(315..1990)
2041462 = sum(414..2062)
2041863 = sum(418..2063)
2158083 = sum(158..2083)
2244132 = sum(244..2132)
2135549 = sum(554..2139)
2349027 = sum(902..2347)
2883558 = sum(883..2558)
2989637 = sum(989..2637)

When you look at nutty sums in other bases, you’ll find that the number “210” is always triangular and always a nutty sum in bases > 2:

210 = sum(1..20) in b3 → 21 = sum(1..6) in b10
210 = sum(1..20) in b4 → 36 = sum(1..8) in b10
210 = sum(1..20) in b5 → 55 = sum(1..10) in b10
210 = sum(1..20) in b6 → 78 = sum(1..12) in b10
210 = sum(1..20) in b7 → 105 = sum(1..14) in b10
210 = sum(1..20) in b8 → 136 = sum(1..16) in b10
210 = sum(1..20) in b9 → 171 = sum(1..18) in b10
210 = sum(1..20) in b10
210 = sum(1..20) in b11 → 253 = sum(1..22) in b10
210 = sum(1..20) in b12 → 300 = sum(1..24) in b10
210 = sum(1..20) in b13 → 351 = sum(1..26) in b10
210 = sum(1..20) in b14 → 406 = sum(1..28) in b10
210 = sum(1..20) in b15 → 465 = sum(1..30) in b10
210 = sum(1..20) in b16 → 528 = sum(1..32) in b10
210 = sum(1..20) in b17 → 595 = sum(1..34) in b10
210 = sum(1..20) in b18 → 666 = sum(1..36) in b10
210 = sum(1..20) in b19 → 741 = sum(1..38) in b10
210 = sum(1..20) in b20 → 820 = sum(1..40) in b10
[…]

Why is 210 always a nutty sum like that? Because the formula for sum(n₁..n₂) is (n₁*n₂) * (n₂-n₁+1) / 2. In all bases > 2, the sum of 1 to 20 (where 20 = 2 * b) is therefore:

(1+20) * (20-1+1) / 2 = 21 * 20 / 2 = 21 * 10 = 210

And here are nutty sums of both kinds (n₁ inside n₂ and n₂ inside n₁) for base 8:

210 = sum(1..20) in b8 → 136 = sum(1..16) in b10
12653 = sum(26..153) → 5547 = sum(22..107)
23711 = sum(71..231) → 10185 = sum(57..153)
2022323 = sum(223..2023) → 533715 = sum(147..1043)
2032472 = sum(247..2032) → 537914 = sum(167..1050)
2271564 = sum(715..2264) → 619380 = sum(461..1204)
2307422 = sum(742..2302) → 626450 = sum(482..1218)
125265253 = sum(2526..15253) → 22375083 = sum(1366..6827)

3246710 = sum(310..2467) in b8 → 871880 = sum(200..1335)
in b10
5326512 = sum(512..3265) → 1420618 = sum(330..1717)
15540671 = sum(1571..5406) → 3588537 = sum(889..2822)
21625720 = sum(2120..6257) → 4664272 = sum(1104..3247)

And for base 9:

125 = sum(2..15) in b9 → 104 = sum(2..14) in b10
210 = sum(1..20) → 171 = sum(1..18)
12858 = sum(28..158) → 8720 = sum(26..134)
1128462 = sum(128..1462) → 609824 = sum(107..1109)
1288588 = sum(288..1588) → 708344 = sum(242..1214)
1475745 = sum(475..1745) → 817817 = sum(392..1337)
2010707 = sum(107..2007) → 1070017 = sum(88..1465)
2034446 = sum(344..2046) → 1085847 = sum(283..1500)
2040258 = sum(402..2058) → 1089341 = sum(326..1511)
2063410 = sum(341..2060) → 1104768 = sum(280..1512)
2215115 = sum(215..2115) → 1191281 = sum(176..1553)
2255505 = sum(555..2205) → 1217840 = sum(455..1625)
2475275 = sum(475..2275) → 1348880 = sum(392..1688)
2735455 = sum(735..2455) → 1499927 = sum(599..1832)

1555 = sum(15..55) in b9 → 1184 = sum(14..50) in b10
155858 = sum(158..558) → 96200 = sum(134..458)
1148181 = sum(181..1481) → 622720 = sum(154..1126)
2211313 = sum(213..2113) → 1188525 = sum(174..1551)
2211747 = sum(247..2117) → 1188880 = sum(205..1555)
6358585 = sum(685..3585) → 3404912 = sum(563..2669)
7037453 = sum(703..3745) → 3745245 = sum(570..2795)
7385484 = sum(784..3854) → 3953767 = sum(643..2884)
13518167 = sum(1367..5181) → 6685072 = sum(1033..3799)
15588588 = sum(1588..5588) → 7794224 = sum(1214..4130)
17603404 = sum(1704..6034) → 8859865 = sum(1300..4405)
26750767 = sum(2667..7507) → 13201360 = sum(2005..5515)

Post-Performative Post-Scriptum…

Viz ’s Mr Logic would be a fan of nutty sums. And unlike real nuts, they wouldn’t prove fatal:

Mr Logic Goes Nuts (strip from Viz comic)

(click for full-size)

Square Pairs

Wednesday, 13th Sep, 2023 by Krilling for Company in Integer Sequences, Mathematics, Powers, Prime numbers and tagged 13, 4n + 1, 4n + 3, Albert Girard, David Wells, facts about 13, mathematics, maths, Penguin Dictionary of Curious and Interesting Mathematics, Pierre de Fermat, primes, Pythagorean primes, sum of two squares, thirteen | Leave a comment

Girard knew and Fermat a few years later proved the beautiful theorem that every prime of the form 4n + 1; that is, the primes 5, 13, 17, 29, 37, 41, 53… is the sum of two squares in exactly one way. Primes of the form 4n + 3, such as 3, 7, 11, 19, 23, 31, 43, 47… are never the sum of two squares. — David Wells, The Penguin Dictionary of Curious and Interesting Numbers (1986), entry for “13”.

Elsewhere other-accessible…

• Fermat’s theorem on sums of two squares
• Pythagorean primes

Bi-Bell Basics

Friday, 8th Sep, 2023 by Krilling for Company in Arithmetic, Base Behavior, Binary numbers, Blancmange Curves, Fibonacci Sequence, Fractals, Geometry, Mathematics, Powers and tagged animated gifs, base 10, base 2, base 3, base 5, base ten, bell curves, binary numbers, digit-sums, Fibonacci numbers, graphs, powers of 2, quinary numbers, recreational math, recreational mathematics, recreational maths, sum of column values, ternary numbers | Leave a comment

Here’s what you might call a Sisyphean sequence. It struggles upward, then slips back, over and over again:

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 1, 2, 2, 3, 2, 3, 3, 4, 2...

The struggle goes on for ever. Every time it reaches a new maximum, it will fall back to 1 at the next step. And in fact 1, 2, 3 and all other integers occur infinitely often in the sequence, because it represents the digit-sums of binary numbers:

1 ← 1 1 = 1+0 ← 10 in binary = 2 in base ten 2 = 1+1 ← 11 = 3 1 = 1+0+0 ← 100 = 4 2 = 1+0+1 ← 101 = 5 2 = 1+1+0 ← 110 = 6 3 = 1+1+1 ← 111 = 7 1 = 1+0+0+0 ← 1000 = 8 2 = 1+0+0+1 ← 1001 = 9 2 = 1+0+1+0 ← 1010 = 10 3 = 1+0+1+1 ← 1011 = 11 2 = 1+1+0+0 ← 1100 = 12 3 = 1+1+0+1 ← 1101 = 13 3 = 1+1+1+0 ← 1110 = 14 4 = 1+1+1+1 ← 1111 = 15 1 = 1+0+0+0+0 ← 10000 = 16 2 = 1+0+0+0+1 ← 10001 = 17 2 = 1+0+0+1+0 ← 10010 = 18 3 = 1+0+0+1+1 ← 10011 = 19 2 = 1+0+1+0+0 ← 10100 = 20

Now here’s a related sequence in which all integers do not occur infinitely often:

1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 11, 12, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 18, 19, 20, 21, 7, 8, 9, 10, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16, 17, 12, 13, 14, 15, 15, 16, 17, 18, 16, 17, 18, 19, 19, 20, 21, 22, 13, 14, 15, 16, 16, 17, 18, 19, 17, 18, 19, 20, 20, 21, 22, 23, 18, 19, 20, 21, 21, 22, 23, 24, 22, 23, 24, 25, 25, 26, 27, 28, 8, 9, 10, 11, 11, 12, 13, 14, 12, 13, 14, 15, 15...

The sequence represents the sum of the values of occupied columns in the binary numbers, reading from right to left:

10 in binary = 2 in base ten 21 (column values from right to left) 2*1 + 1*0 = 2 11 = 3 21 2*1 + 1*1 = 3 100 = 4 321 (column values from right to left) 3*1 + 2*0 + 1*0 = 3 101 = 5 321 3*1 + 2*0 + 1*1 = 4 110 = 6 321 3*1 + 2*1 + 1*0 = 5 111 = 7 321 3*1 + 2*1 + 1*1 = 6 1000 = 8 4321 4*1 + 3*0 + 2*0 + 1*0 = 4 1001 = 9 4321 4*1 + 3*0 + 2*0 + 1*1 = 5 1010 = 10 4321 4*1 + 3*0 + 2*1 + 1*0 = 6 1011 = 11 4321 4*1 + 3*0 + 2*1 + 1*1 = 7 1100 = 12 4321 4*1 + 3*1 + 2*0 + 1*0 = 7 1101 = 13 4321 4*1 + 3*1 + 2*0 + 1*1 = 8 1110 = 14 4321 4*1 + 3*1 + 2*1 + 1*0 = 9 1111 = 15 4321 4*1 + 3*1 + 2*1 + 1*1 = 10 10000 = 16 54321 5*1 + 4*0 + 3*0 + 2*0 + 1*0 = 5

In that sequence, although no number occurs infinitely often, some numbers occur more often than others. If you represent the count of sums up to a certain digit-length as a graph, you get a famous shape:

Bell curve formed by the count of column-sums in base 2

Bi-bell curves for 1 to 16 binary digits (animated)

In “Pi in the Bi”, I looked at that way of forming the bell curve and called it the bi-bell curve. Now I want to go further. Suppose that you assign varying values to the columns and try other bases. For example, what happens if you assign the values 2^p + 1 to the columns, reading from right to left, then use base 3 to generate the sums? These are the values of 2^p + 1:

2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025...

And here’s an example of how you generate a column-sum in base 3:

2102 in base 3 = 65 in base ten 9532 (column values from right to left) 2*9 + 1*5 + 0*3 + 2*2 = 27

The graphs for these column-sums using base 3 look like this as the digit-length rises. They’re no longer bell-curves (and please note that widths and heights have been normalized so that all graphs fit the same space):

Graph for the count of column-sums in base 3 using 2^p + 1 (digit-length <= 7)

(width and height are normalized)

Graph for base 3 and 2^p + 1 (dl<=8)

Graph for base 3 and 2^p + 1 (dl<=9)

Graph for base 3 and 2^p + 1 (dl<=10)

Graph for base 3 and 2^p + 1 (dl<=11)

Graph for base 3 and 2^p + 1 (dl<=12)

Graph for base 3 and 2^p + 1 (animated)

Now try base 3 and column-values of 2^p + 2 = 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026…

Graph for base 3 and 2^p + 2 (dl<=7)

Graph for base 3 and 2^p + 2 (dl<=8)

Graph for base 3 and 2^p + 2 (dl<=9)

Graph for base 3 and 2^p + 2 (dl<=10)

Graph for base 3 and 2^p + 2 (animated)

Now try base 5 and 2^p + 1 for the columns. The original bell curve has become like a fractal called the blancmange curve:

Graph for base 5 and 2^p + 1 (dl<=7)

Graph for base 5 and 2^p + 1 (dl<=8)

Graph for base 5 and 2^p + 1 (dl<=9)

Graph for base 5 and 2^p + 1 (dl<=10)

Graph for base 5 and 2^p + 1 (animated)

And finally, return to base 2 and try the Fibonacci numbers for the columns:

Graph for base 2 and Fibonacci numbers = 1,1,2,3,5… (dl<=7)

Graph for base 2 and Fibonacci numbers (dl<=9)

Graph for base 2 and Fibonacci numbers (dl<=11)

Graph for base 2 and Fibonacci numbers (dl<=13)

Graph for base 2 and Fibonacci numbers (dl<=15)

Graph for base 2 and Fibonacci numbers (animated)

Previously Pre-Posted…

• Pi in the Bi — bell curves generated by binary digits

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

მეფე (2) • მსოფლიოს (3) • მათემატიკა (5) •

Category Archives: Mathematics

Nexcelsior

Pirouetting the Perimeter

Grow Fourth

Back to LIFE

Eternal LIFE

Pyramidic Palindromes

Blancmange Butterfly

Nuts for Numbers

Square Pairs

Bi-Bell Basics