A255488 Number of odd terms in expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.

1, 6, 6, 12, 6, 16, 12, 24, 6, 36, 16, 32, 12, 36, 24, 48, 6, 36, 36, 72, 16, 56, 32, 64, 12, 72, 36, 72, 24, 68, 48, 96, 6, 36, 36, 72, 36, 96, 72, 144, 16, 96, 56, 112, 32, 100, 64, 128, 12, 72, 72, 144, 36, 120, 72, 144, 24, 144, 68, 136

Offset

0,2

Comments

All the following are of the same type: A001316, A071053, A134660, A134661, A134662, A255485, A247649, A255486. It would be nice to have some unifying formula or recurrence. (Restating the definition, these are the Hamming weights of the n-th powers of the corresponding polynomials over GF(2). - Joerg Arndt, Mar 02 2015)

Links

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

Example

From Omar E. Pol, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
6;
6,12;
6,16,12,24;
6,36,16,32,12,36,24,48;
6,36,36,72,16,56,32,64,12,72,36,72,24,68,48,96;
6,36,36,72,36,96,72,144,16,96,56,112,32,100,64,128,12,72,72,144,36,120,72,144,24,144,68,136...
...
In each row the first quarter of the terms (and no more) are equal to 6 times the beginning of the sequence itself (corrected after Sloane's comment in A247649, Mar 03 2015).
(End)

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^3);

Mathematica

a[n_] := Count[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], _?OddQ];
Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Apr 06 2017 *)

Prog

(PARI) a(n) = {my(pol=(1+x+x^2+x^3+x^4+x^5)*Mod(1, 2)); subst(lift(pol^n), x, 1); } \\ Michel Marcus, Mar 01 2015

Crossrefs

Cf. A001316, A071053, A134660, A134661, A134662, A255485, A247649, A255486.

Sequence in context: A040031 A205695 A205029 * A247513 A124508 A028317

Adjacent sequences: A255485 A255486 A255487 * A255489 A255490 A255491

Keyword

nonn

Author

N. J. A. Sloane, Mar 01 2015

Status

approved