A170838 G.f.: Product_{k>=0} (1 + x^(2^k-1) + 3x^(2^k)).

2, 5, 9, 11, 11, 24, 36, 29, 11, 24, 38, 44, 57, 108, 135, 83, 11, 24, 38, 44, 57, 108, 137, 98, 57, 110, 158, 189, 279, 459, 486, 245, 11, 24, 38, 44, 57, 108, 137, 98, 57, 110, 158, 189, 279, 459, 488, 260, 57, 110, 158, 189, 279, 461, 509, 351, 281, 488, 663, 846, 1296

Offset: 0

Author

N. J. A. Sloane, Jan 02 2010

Keywords

• nonn

Links

• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

A170838-A170852, A170854-A170872 were added to supplement Gary W. Adamson's A162956.

Programs

Maple

Maple program for A170838-A170852, A162956, A170854-A170872.
read format;
G := proc(a,b,c); mul( 1 + a*x^(2^n-1) + b*x^(2^n), n=c..20); end;
f := proc(a,b,c) seriestolist(series(G(a,b,c),x,120)); end;
at:=170838:
for a from 1 to 2 do for c from 0 to 2 do
b:=3;
t1:=f(a,b,c);
lprint( format(t1,at) );
lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
at:=at+1; od: od:
for b from 1 to 3 do for c from 0 to 2 do
a:=3;
t1:=f(a,b,c);
lprint( format(t1,at) );
lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
at:=at+1; od: od:
h:=proc(r,s,a,b) local s1,n,i,j;
s1:=array(0..120);
s1[0]:=r; s1[1]:=s;
for n from 2 to 120 do i:=floor(log(n)/log(2));
j:=n-2^i; s1[n]:=a*s1[j]+b*s1[j+1]; od:
[seq(s1[n],n=0..120)];
end;
l1:=[[0,1],[1,0],[1,1],[1,2]];
l2:=[[3,1],[3,2],[1,3],[2,3],[3,3]];
for i from 1 to 4 do for j from 1 to 5 do
r:=l1[i][1];
s:=l1[i][2];
a:=l2[j][1];
b:=l2[j][2];
t1:=h(r,s,a,b);
lprint(format(t1,at)); at:=at+1;
lprint("a(0)=",r,", a(1)=", s, "; a(2^i+j)=",a,"*a(j)+",b,"a(j+1) for 0 <= j < 2^i.");
od: od:

Mathematica

With[{nn=60},CoefficientList[Series[Product[1+x^(2^k-1)+3x^2^k,{k,0,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Dec 29 2021 *)