A255485 Number of odd terms in expansion of (1 + x + x^2 + x^4)^n.

1, 4, 4, 8, 4, 12, 8, 14, 4, 16, 12, 24, 8, 24, 14, 30, 4, 16, 16, 32, 12, 36, 24, 44, 8, 32, 24, 48, 14, 46, 30, 60, 4, 16, 16, 32, 16, 48, 32, 56, 12, 48, 36, 72, 24, 76, 44, 92, 8, 32, 32, 64, 24, 72, 48, 88, 14, 56, 46, 92, 30, 96, 60, 118, 4, 16, 16, 32, 16, 48, 32, 56, 16, 64, 48, 96

Offset: 0

Author

N. J. A. Sloane, Feb 28 2015

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001316, A071053, A134660, A134661, A134662, A255486.

Programs

Maple

r1:=proc(f) local g,n; g:=n->nops(expand(f^n) mod 2); [seq(g(n),n=0..90)]; end;
r1(1+x+x^2+x^4);
# Alternative:
P:= 1:
for n from 0 to 100 do
  A[n]:= nops(P);
  P:= expand(P*(1+x+x^2+x^4)) mod 2;
od:
seq(A[i],i=0..100); # Robert Israel, Jan 07 2018

Mathematica

a[n_] := Count[(List @@ Expand[(1+x+x^2+x^4)^n]) /. x -> 1, _?OddQ]; a[0] = 1;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 06 2023 *)

PARI

a(n) = {my(pol=(1+x+x^2+x^4)*Mod(1,2)); subst(lift(pol^n), x, 1);} \\ Michel Marcus, Mar 01 2015

Formula

From Robert Israel, Jan 07 2018: (Start)

a(2*n) = a(n).

a(8*n+1) = 4*a(n). (End)