A265056 Partial sums of A234275.

1, 5, 21, 45, 77, 117, 165, 221, 285, 357, 437, 525, 621, 725, 837, 957, 1085, 1221, 1365, 1517, 1677, 1845, 2021, 2205, 2397, 2597, 2805, 3021, 3245, 3477, 3717, 3965, 4221, 4485, 4757, 5037, 5325, 5621, 5925, 6237, 6557, 6885, 7221, 7565, 7917, 8277, 8645, 9021, 9405, 9797, 10197, 10605, 11021

Offset: 0

Author

N. J. A. Sloane, Dec 28 2015

Keywords

• nonn
• easy

Comments

The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood, initiated with a single black (ON) cell.- Robert Price, May 28 2016

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to 2D 5-Neighbor Cellular Automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A234275.

Apart from initial term, same as A078371.

Programs

Mathematica

Accumulate[LinearRecurrence[{2,-1},{1,4,16,24},60]] (* or *) LinearRecurrence[{3,-3,1},{1,5,21,45},60] (* Harvey P. Dale, Sep 22 2024 *)

PARI

Vec((1+2*x+9*x^2-4*x^3)/(1-x)^3 + O(x^100)) \\ Colin Barker, Jan 01 2016

Formula

From Colin Barker, Jan 01 2016: (Start)

a(n) = 4*n^2+4*n-3 for n>0.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.

G.f.: (1+2*x+9*x^2-4*x^3) / (1-x)^3.

(End)