A253067 -id:A253067 - cp's OEIS frontend

A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y.

1, 5, 5, 17, 5, 25, 17, 65, 5, 25, 25, 85, 17, 85, 65, 229, 5, 25, 25, 85, 25, 125, 85, 325, 17, 85, 85, 289, 65, 325, 229, 813, 5, 25, 25, 85, 25, 125, 85, 325, 25, 125, 125, 425, 85, 425, 325, 1145, 17, 85, 85, 289, 85, 425, 289, 1105, 65, 325, 325, 1105, 229, 1145, 813, 2945, 5, 25, 25, 85

Offset: 0

N. J. A. Sloane, Jan 26 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 171 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[0, 0, X]
[X, X, X]
[0, 0, X]
which contains a(1) = 5 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
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

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253066.

Cf. A253067.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1+x+x^2+x^2*y+x^2/y;
OddCA(f, 130);

(* f = A253067 *) f[0]=1; f[1]=5; f[2]=17; f[3]=65; f[4]=229; f[5]=813; f[n_] := f[n] = 8 f[n-5] + 6 f[n-4] + 13 f[n-3] + 5 f[n-2] + f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 67}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253067.

A253070 The subsequence A253069(2^n-1).

Original entry on oeis.org

1, 6, 22, 82, 302, 1106, 4066, 14902, 54678, 200578, 735770, 2699182, 9901550, 36323050, 133247570, 488805718, 1793137798, 6577952882, 24130592458, 88520767614, 324729961566, 1191240790586, 4369952806274, 16030753627238, 58807285300086, 215728897446594, 791380812129402, 2903104763095054

Offset: 0

N. J. A. Sloane, Jan 29 2015

A253069 is the Run Length Transform of this sequence.

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 (3,3,-1,-6,10,-8,8).

Cf. A253067, A253068, A253069.

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;
f:=1/x+1+x+x/y+y/x+x*y;
OddCA2(f,10);

LinearRecurrence[{3, 3, -1, -6, 10, -8, 8}, {1, 6, 22, 82, 302, 1106, 4066}, 28] (* Jean-François Alcover, Nov 27 2017 *)

Vec(-(2*x+1)*(2*x^5+x^3-x^2+x+1)/(8*x^7-8*x^6+10*x^5-6*x^4-x^3+3*x^2+3*x-1) + O(x^30)) \\ Colin Barker, Jul 16 2015

G.f.: (1+2*x)*(1+x-x^2+x^3+2*x^5)/(1-3*x-3*x^2+x^3+6*x^4-10*x^5+8*x^6-8*x^7). - Doron Zeilberger, Feb 18 2015

a(11) and a(12) (Maple on a 32 GB machine) from R. J. Mathar, Feb 04 2015

a(13) onwards from Doron Zeilberger, Feb 18 2015 (the terms previously listed were wrong). - N. J. A. Sloane, Feb 19 2015

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

Original entry on oeis.org

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

N. J. A. Sloane, Jan 31 2015

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

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).

Cf. A253067, A253068, A253071, A050476.

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);

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

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

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).

cp's OEIS Frontend

A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y.

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A253070 The subsequence A253069(2^n-1).

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

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