A253072 The subsequence A253071(2^n-1).

1, 7, 21, 95, 333, 1319, 4837, 18447, 68733, 259447, 972565, 3661535, 13756333, 51754567, 194586181, 731919279, 2752461533, 10352254743, 38932913525, 146424889471, 550683608589, 2071066796007, 7789015542949, 29293584500047, 110169505843517, 414334209685687

Offset: 0

Author

N. J. A. Sloane, Jan 31 2015

Keywords

• nonn
• easy

Comments

A253071 is the Run Length Transform of this sequence.

A253072(2^k-1) = A050476(2^k-1), 0<=k<=3. This is just a coincidence, since it fails at m=4. - Omar E. Pol, Feb 01 2015; N. J. A. Sloane, Feb 20 2015

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, 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
• Index entries for linear recurrences with constant coefficients, signature (6,-5,-24,44,-8).

Crossrefs

Cf. A253067, A253068, A253071, A050476.

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;
f25:=1/(x*y)+1/x+1/y+y+x/y+x+x*y;
OddCA2(f25,8);

Mathematica

LinearRecurrence[{6, -5, -24, 44, -8}, {1, 7, 21, 95, 333}, 26] (* Jean-François Alcover, Nov 27 2017 *)

PARI

Vec(-(8*x^4-28*x^3+16*x^2-x-1)/(8*x^5-44*x^4+24*x^3+5*x^2-6*x+1) + O(x^30)) \\ Colin Barker, Jul 16 2015

Formula

G.f.: -(-1-t+16*t^2-28*t^3+8*t^4)/(1-6*t+5*t^2+24*t^3-44*t^4+8*t^5).