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

0,2

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

For n > 0, a(n) is the number of unit squares visible when two (2n-1) X (2n+1) grids are placed perpendicular to each other with their centers aligned. - Kiran Ananthpur Bacche, Sep 04 2025

References

Stephen 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.

Stephen 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).

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)

From Elmo R. Oliveira, Sep 08 2025: (Start)

E.g.f.: 4 + (4*x^2 + 8*x - 3)*exp(x).

a(n) = A060747(n)*A005408(n+1) for n > 0. (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A005408, A060747, A234275.

Apart from initial term, same as A078371.

Sequence in context: A031292 A147331 A078371 * A049741 A166010 A146846

Adjacent sequences: A265053 A265054 A265055 * A265057 A265058 A265059

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Dec 28 2015

Status

approved