cp's OEIS Frontend

A256250 Total number of ON states after n generations of a cellular automaton on the square grid.

%I A256250 #30 Nov 03 2022 08:43:16
%S A256250 1,5,9,21,25,37,57,85,89,101,121,149,185,229,281,341,345,357,377,405,
%T A256250 441,485,537,597,665,741,825,917,1017,1125,1241,1365,1369,1381,1401,
%U A256250 1429,1465,1509,1561,1621,1689,1765,1849,1941,2041,2149,2265,2389,2521,2661,2809,2965,3129,3301,3481,3669,3865,4069,4281,4501,4729,4965,5209,5461
%N A256250 Total number of ON states after n generations of a cellular automaton on the square grid.
%C A256250 A256251 gives the number of cells turned ON at n-th stage.
%C A256250 Note that the number of cells turned ON at n-th stage in each one of its four quadrants is also A006257 (Josephus problem). For more information see A256249.
%C A256250 It appears that this is also a bisection of A256249.
%C A256250 First differs from A169707 at a(13), but both sequences share infinitely many terms. This one is simpler. Compare A169707.
%H A256250 Michael De Vlieger, <a href="/A256250/b256250.txt">Table of n, a(n) for n = 1..16384</a>
%H A256250 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. 37.
%H A256250 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 A256250 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A256250 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F A256250 a(n) = 1 + 4*A256249(n-1), n >= 1.
%e A256250 Also, written as an irregular triangle T(n,k), k >= 1, in which the row lengths are the terms of A011782 the sequence begins:
%e A256250 1;
%e A256250 5;
%e A256250 9,   21;
%e A256250 25,  37, 57, 85;
%e A256250 89, 101,121,149,185,229,281,341;
%e A256250 345,357,377,405,441,485,537,597,665,741,825,917,1017,1125,1241,1365;
%e A256250 ...
%e A256250 Right border gives the positive terms of A002450.
%e A256250 It appears that this triangle at least shares with the triangles from the following sequences; A147562, A162795, A169707, A255366, the positive elements of the columns k, if k is a power of 2.
%t A256250 1 + 4*Accumulate@ Prepend[Flatten@ Table[Range[1, 2^n - 1, 2], {n, 0, 7}], 0] (* _Michael De Vlieger_, Nov 03 2022, after _Ivan N. Ianakiev_ at A256249 *)
%Y A256250 Cf. A002450, A006257, A011782, A139250, A147562, A160720, A162795, A169707, A246335, A255366, A256138, A256249, A256251.
%K A256250 nonn,look
%O A256250 1,2
%A A256250 _Omar E. Pol_, Mar 20 2015