A255471 -id:A255471 - cp's OEIS frontend

A255470 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 176 when started with a single ON cell.

1, 6, 6, 24, 6, 36, 24, 100, 6, 36, 36, 144, 24, 144, 100, 396, 6, 36, 36, 144, 36, 216, 144, 600, 24, 144, 144, 576, 100, 600, 396, 1596, 6, 36, 36, 144, 36, 216, 144, 600, 36, 216, 216, 864, 144, 864, 600, 2376, 24, 144, 144, 576, 144, 864, 576, 2400, 100, 600, 600, 2400

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 23 2015

nonn

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Run length transform of A255471.

A370627 a(n) = 2^(n - 1)((-1)^(n + 1) + 72^n)/3 = 2^(n - 1)*A062092(n).

Original entry on oeis.org

1, 5, 18, 76, 296, 1200, 4768, 19136, 76416, 305920, 1223168, 4893696, 19572736, 78295040, 313171968, 1252704256, 5010784256, 20043202560, 80172679168, 320690978816, 1282763390976, 5131054612480, 20524216352768, 82096869605376, 328387470032896, 1313549896908800, 5254199554080768

Offset: 0

Paul Curtz, Jul 03 2024

Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003683, A122803, A133125, A255470, A255471.

Cf. A108982, A062092.

LinearRecurrence[{2, 8}, {1, 5}, 27] (* Amiram Eldar, Jul 03 2024 *)

a(n) = 2^(n-1)*((-1)^(n+1) + 7*2^n)/3 \\ Thomas Scheuerle, Jul 03 2024

Binomial transform of A133125.
G.f.: (1 + 3*x)/(1 - 2*x - 8*x^2).
E.g.f.: (1/3)*exp(x)*(3*exp(3*x) + sinh(3*x)).
a(n) = 2*a(n-1) + 8*a(n-2), for n > 1.
a(n) = 4*a(n-1) + (-2)^n, for n > 0.
a(n) = (a(n+2) - 2*a(n+1))/8.
From Thomas Scheuerle, Jul 03 2024: (Start)
a(n) = 2^(n - 1)*((-1)^(n + 1) + 7*2^n)/3.
a(n) = A003683(n) + 4^n.
a(n) = A255470(2^n - 1) - A255470(2^(n-1) - 1) = A255471(n) - A255471(n-1), for n > 0. (End)
Binomial transform: A108982.

cp's OEIS Frontend

A255470 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 176 when started with a single ON cell.

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A370627 a(n) = 2^(n - 1)((-1)^(n + 1) + 72^n)/3 = 2^(n - 1)*A062092(n).

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Author

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Mathematica

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Formula

cp's OEIS Frontend

A255470 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 176 when started with a single ON cell.

Views

Author

Keywords

Links

Crossrefs

A370627 a(n) = 2^(n - 1)*((-1)^(n + 1) + 7*2^n)/3 = 2^(n - 1)*A062092(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370627 a(n) = 2^(n - 1)((-1)^(n + 1) + 72^n)/3 = 2^(n - 1)*A062092(n).