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

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 149, 185, 229, 281, 341, 345, 357, 377, 405, 441, 485, 537, 597, 665, 741, 825, 917, 1017, 1125, 1241, 1365, 1369, 1381, 1401, 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

Offset: 1

Author

Omar E. Pol, Mar 20 2015

Keywords

• nonn
• look

Comments

A256251 gives the number of cells turned ON at n-th stage.

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.

It appears that this is also a bisection of A256249.

First differs from A169707 at a(13), but both sequences share infinitely many terms. This one is simpler. Compare A169707.

Examples

			Also, written as an irregular triangle T(n,k), k >= 1, in which the row lengths are the terms of A011782 the sequence begins:
1;
5;
9,   21;
25,  37, 57, 85;
89, 101,121,149,185,229,281,341;
345,357,377,405,441,485,537,597,665,741,825,917,1017,1125,1241,1365;
...
Right border gives the positive terms of A002450.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A002450, A006257, A011782, A139250, A147562, A160720, A162795, A169707, A246335, A255366, A256138, A256249, A256251.

Programs

Mathematica

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 *)

Formula

a(n) = 1 + 4*A256249(n-1), n >= 1.