A169780 -id:A169780 - cp's OEIS frontend

A169781 a(n) = A169780(2^n).

1, 3, 9, 29, 99, 361, 1373, 5349, 21109, 83859, 334275, 1334769, 5334413, 21328351, 85294869

Offset: 0

N. J. A. Sloane, May 11 2010

These are the "high water marks" in A169780.

initTriangle[ ] := {{1, 0, 1, 0, 1}, {1, 0, 1, 1, 0, 1}, {1, 0, 1, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}};
nextRow[k_] := Join[{1, 0}, Table[0, k], {0, 1}]
neighborPosList[{i_, j_}] := {{i - 1, j - 1}, {i - 1, j}, {i, j - 1}, {i, j + 1}, {i + 1, j}, {i + 1, j + 1}}
neighborEmptyList[{i_, j_}] := Select[neighborPosList[{i, j}], nTriangle[[Apply[Sequence, #]]] == 0 &]
getCellValue[{i_, j_}] := nTriangle[[i, j]]
a169781[n_] := Module[{newGen = { }, lastGen = {{5, 3}, {5, 7}}, newCount = 0, totalCount = 11, k, p, emptyNs, q, emptyCs, r}, nTriangle = initTriangle[ ]; For[k = 6, k <= n, k++, AppendTo[nTriangle, nextRow[k + 1]]; For[p = 1, p <= Length[lastGen], p++, emptyNs = Select[neighborPosList[lastGen[[p]]], getCellValue[#] == 0 &]; For[q = 1, q <= Length[emptyNs], q++, emptyCs = Select[neighborPosList[emptyNs[[q]]], getCellValue[#] == 1 &]; If[Length[emptyCs] == 1, AppendTo[newGen, emptyNs[[q]]]]]]; For[r = 1, r <= Length[newGen], r++, nTriangle[[Apply[Sequence, newGen[[r]]]]] = 1]; lastGen = newGen; totalCount += Length[newGen]; newGen = { }]; totalCount]/;(n>=2^3 && IntegerQ[Log[2, n]])
a169781[2^11] (* sample invocation for a(11) - Hartmut F. W. Hoft, Apr 17 2020 *)

a(11)-a(14) from Hartmut F. W. Hoft, Apr 17 2020

A169788 a(n) = n(n+1)/2 - A169780(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 6, 7, 7, 14, 20, 25, 27, 30, 36, 37, 37, 52, 66, 79, 89, 100, 112, 117, 115, 122, 140, 153, 157, 158, 168, 167, 167, 198, 228, 257, 283, 310, 338, 359, 373, 396, 428, 453, 467, 474, 486, 491, 481, 496, 538, 575, 603, 628, 654, 665, 653, 650, 680, 707, 711

Offset: 0

N. J. A. Sloane, May 13 2010

nonn

If n is a power of 2, this is the number of OFF cells after n stages in a 60-degree wedge of the hexagonal CA (see A170905, A169780, A151723, A169789).

N. J. A. Sloane, Table of n, a(n) for n = 0..1025

Cf. A170905, A169780, A151723, A169789.

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 55, 85, 127, 133, 151, 181, 235, 289, 331, 409, 499, 505, 523, 553, 607, 661, 715, 817, 967, 1069, 1111, 1189, 1327, 1489, 1603, 1789, 1975, 1981, 1999, 2029, 2083, 2137, 2191, 2293, 2443, 2545, 2599, 2701, 2875, 3097, 3295

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
It would be nice to find a recurrence for this sequence!
Has a behavior similar to A182840 and possibly to A182632. - Omar E. Pol, Jan 15 2016

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1026 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Bradley Klee, Log-periodic coloring, over the half-hexagon tiling.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

A151723[0] = 0; A151723[n_] := Total[CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151723, 47, 0](* JungHwan Min, Sep 01 2016 *)
A151723L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151723L[46] (* JungHwan Min, Sep 01 2016 *)

a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
a(n) = 1 + 6*A169779(n-2), n >= 2. - Omar E. Pol, Mar 19 2015
It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - Omar E. Pol, Apr 04 2015

Edited by N. J. A. Sloane, Jan 10 2010

A169779 Partial sums of A170898.

Original entry on oeis.org

1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 39, 48, 55, 68, 83, 84, 87, 92, 101, 110, 119, 136, 161, 178, 185, 198, 221, 248, 267, 298, 329, 330, 333, 338, 347, 356, 365, 382, 407, 424, 433, 450, 479, 516, 549, 590, 647, 680, 687, 700, 723, 750, 777, 820, 887, 946, 973, 1004, 1059

Offset: 0

N. J. A. Sloane, May 10 2010

nonn

On the hexagonal grid consider an infinite 60-degree wedge. A cell is turned ON if exactly one of its six neighbors is ON. We start with a single ON cell. An ON cell remains ON forever. The sequence counts the total number of ON states after n generations. The structure is also the tree that arises from one of the six spokes of the structure of A151723. For n >> 1 the structure looks like a quadrilateral formed by two scalene right triangles which are joined at their hypotenuses. - Omar E. Pol, Mar 06 2013

N. J. A. Sloane, Table of n, a(n) for n = 0..1023
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A151723, A169780.

a(n) = (A151723(n+2) - 1)/6. - Omar E. Pol, Mar 06 2013

A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60-degree wedge (including the two bounding edges).

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 26, 18, 8, 14, 24, 28, 20, 32, 32, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42, 58, 34, 8, 14, 24, 28, 28, 44, 68, 60, 28, 32, 56, 70, 50, 70, 64, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42

Offset: 0

N. J. A. Sloane, Jan 22 2010

			From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle starting from 1, the right border gives A000079 and row lengths give A011782.
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,26,18,8,14,24,28,20,32,32;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,58,34,8,14,24,28,28,44,68,60,28,32,56,70,50,70,64;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,...
... (End)

N. J. A. Sloane, Table of n, a(n) for n = 0..1025
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151723, A151724, A170898, A169778, A169780 (partial sums).

a(n) = A170898(n-2) + 1 for n >= 2.

a(n) = 2*A169778(n) for n != 1.

A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 509, 547, 573, 615, 667, 701, 709, 723, 745, 771

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited, except in the substructures in which the symmetry could be broken, so a(44) = 509, not 507. For another version with broken symmetry in some substructures see A233764. See also A231348, a simpler cellular automaton based in triangles which has essentially a similar structure.
A233781 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233764 at a(44).

Cf. A139250, A169780, A220500, A231348, A233764, A233781, A233970.

A233970 Toothpick sequence on hexagonal net starting from the vertex of a 60-degree wedge (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 205, 213, 227, 249, 275, 291, 321, 353, 355, 359, 365, 375, 385, 395, 413, 437, 459, 469, 487, 515, 553, 581, 627, 683, 737, 745, 759, 781, 807

Offset: 0

Omar E. Pol, Dec 18 2013

nonn

Toothpicks are connected by their endpoints. The toothpicks placed in north direction are prohibited. The sequence gives the number of toothpicks after n-th stage in the structure. A233971 (the first differences) give the number of toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from both A233764 and A233780 at a(25).

Cf. A139250, A161206, A161830, A169780, A182632, A182634, A182634, A182840, A233760, A233764, A233780, A233971, A233972.

A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 507, 545, 575, 625, 679, 713, 721, 735, 757, 783

Offset: 0

Omar E. Pol, Dec 16 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited. Note that due this rule there are substructures with broken symmetry, for instance a(44) = 507, not 509. For another version without broken symmetry see A233780.
A233765 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233780 at a(44).

Cf. A139250, A169780, A220500, A220520, A231348, A233765, A233780, A233970.

A256138 Total number of ON states after n generations of cellular automaton of A151723 based on hexagons, if we only look at two opposite 120-degree wedges, including the central cell.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 157, 193, 221, 273, 333, 337, 349, 369, 405, 441, 477, 545, 645, 713, 741, 793, 885, 993, 1069, 1193, 1317, 1321, 1333, 1353, 1389, 1425, 1461, 1529, 1629, 1697, 1733, 1801, 1917, 2065, 2197, 2361, 2589, 2721, 2749, 2801, 2893, 3001, 3109, 3281, 3549, 3785, 3893, 4017, 4237, 4513, 4709, 4985, 5237

Offset: 1

Omar E. Pol, Mar 20 2015

nonn

First differs from both A169707 and A246335 at a(12).
First differs from the average of A169707 and A246335 at a(13).
Note that the above mentioned cellular automata work on the square grid.
A256139 gives the number of cells turned ON at the n-th stage.

Cf. A151723, A169707, A169779, A169780, A170898, A246333, A246335, A256139.

a(n) = 1 + 2*(A151723(n) - 1)/3 = 1 - 4*n + 4*A169780(n).
a(n) = 1 + 4*A169779(n-2), n >= 2.
a(n) = A151723(n) - 2*A169779(n-2), n >= 2.

cp's OEIS Frontend

A169781 a(n) = A169780(2^n).

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A169788 a(n) = n(n+1)/2 - A169780(n).

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A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

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A169779 Partial sums of A170898.

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A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60-degree wedge (including the two bounding edges).

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A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A233970 Toothpick sequence on hexagonal net starting from the vertex of a 60-degree wedge (see Comments lines for precise definition).

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A233764 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A256138 Total number of ON states after n generations of cellular automaton of A151723 based on hexagons, if we only look at two opposite 120-degree wedges, including the central cell.

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