A256256 - OEIS

%I #48 Nov 03 2022 10:07:59

%S 0,6,18,30,54,66,90,114,162,174,198,222,270,294,342,390,486,498,522,

%T 546,594,618,666,714,810,834,882,930,1026,1074,1170,1266,1458,1470,

%U 1494,1518,1566,1590,1638,1686,1782,1806,1854,1902,1998,2046,2142,2238,2430,2454,2502,2550,2646,2694,2790,2886,3078,3126,3222,3318,3510,3606,3798,3990,4374

%N Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

%C Analog of A160720, but here we are working on the triangular lattice.

%C The first differences (A256257) gives the number of triangular cells turned ON at every generation.

%C Also 6 times the sum of all entries in rows 0 to n of Sierpiński's triangle A047999.

%C Also 6 times the total number of odd entries in first n rows of Pascal's triangle A007318, see formula section.

%C The structure contains three distinct kinds of polygons formed by triangular ON cells: the initial hexagon, rhombuses (each one formed by two ON cells) and unit triangles.

%C Note that if n is a power of 2 greater than 2, the structure looks like concentric hexagons with triangular holes, where some of them form concentric stars.

%H Michael De Vlieger, <a href="/A256256/b256256.txt">Table of n, a(n) for n = 0..16386</a>

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 30.

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = 6*A006046(n).

%e On the infinite triangular grid we start with all triangular cells turned OFF, so a(0) = 0.

%e At stage 1, in the structure there are six triangular cells turned ON forming a regular hexagon, so a(1) = 6.

%e At stage 2, there are 12 new triangular ON cells forming six rhombuses around the initial hexagon, so a(2) = 6 + 12 = 18.

%e And so on.

%t Prepend[6*FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]], 0] &[63] (* _Michael De Vlieger_, Nov 03 2022, after _Bradley Klee_ at A006046 *)

%o (PARI) a(n) = 6*sum(j=0, n, sum(k=0, j, binomial(j, k) % 2)); \\ _Michel Marcus_, Apr 01 2015

%Y Cf. A001316, A006046, A007318, A025192, A047999, A151723, A160120, A160720, A161644, A256257.

%K nonn

%O 0,2

%A _Omar E. Pol_, Mar 20 2015