A253067 The subsequence A253065(2^n-1).

1, 5, 17, 65, 229, 813, 2945, 10513, 37701, 135261, 484609, 1737665, 6229413, 22330829, 80057281, 286996657, 1028861637, 3688409853, 13222664897, 47402353633, 169934149285, 609201913325, 2183946525185, 7829295473489, 28067476697413, 100619943566493

Offset: 0

Author

N. J. A. Sloane, Jan 27 2015

Keywords

• nonn
• easy

Comments

A253065 is the Run Length Transform of this sequence.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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 [math.CO], 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 [math.CO], 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 [math.CO], 2015.
• Index entries for sequences related to cellular automata
• Index entries for linear recurrences with constant coefficients, signature (1,5,13,6,8).

Crossrefs

Cf. A253065.

Programs

Maple

OddCA2:=proc(f,M) local n,a,i,f2,g,p;
   f2:=simplify(expand(f)) mod 2;
   p:=1; g:=f2;
   for n from 1 to M do p:=expand(p*g) mod 2; print(n,nops(p)); g:=expand(g^2) mod 2; od:
   return;
end;
f24:=1/x+1+x+x/y+x*y;
OddCA2(f24,8);

Mathematica

LinearRecurrence[{1, 5, 13, 6, 8}, {1, 5, 17, 65, 229}, 26] (* Jean-François Alcover, Nov 23 2017 *)

PARI

Vec(-(2*x+1)*(4*x^3+3*x^2+2*x+1)/(8*x^5+6*x^4+13*x^3+5*x^2+x-1) + O(x^30)) \\ Colin Barker, Jul 16 2015

Formula

G.f.: (1+2*x)*(1+2*x+3*x^2+4*x^3)/(1-x-5*x^2-13*x^3-6*x^4-8*x^5).

Examination of the roots of the denominator shows that the ratio of successive terms approaches 3.5849301...