A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.

1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 4, 8, 4, 16, 16, 8, 8, 32, 16, 32, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 8, 16, 8, 32, 32, 16, 16, 64, 32, 64, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 16, 16, 64

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Formula

a(n) = 2^A036555(n).

a(n) = gcd(4^n, C(4*n, n)). - Peter Luschny, Nov 08 2011

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;
4;
4,4;
4,16,4,8;
4,16,16,4,4,16,8,16;
4,16,16,16,16,64,4,8,4,16,16,8,8,32,16,32;
4,16,16,16,16,64,16,32,16,64,64,4,4,16,8,16,4,16,16,16,16,64,8,16,8,32,32,16,16,64,32,64;
...
(End)

Maple

seq(igcd(4^n, binomial(4*n, n)), n=0..77); # Peter Luschny, Nov 08 2011

Mathematica

PolynomialMod[(1+x+x^2+x^3)^n, 2] /. x->1
A036555 = Total /@ IntegerDigits[3 Range[0, 100], 2]; Table[2^A036555[[n]], {n, 1, 20}] (* or *) Table[GCD[4^n, Binomial[4*n, n]], {n, 0, 50}] (* G. C. Greubel, Dec 31 2017 *)

Prog

(PARI) a(n) = {my(pol= Pol([1, 1, 1, 1], xx)*Mod(1, 2)); subst(lift(pol^n), xx, 1); } \\ Michel Marcus, Mar 01 2015
(PARI) a(n) = 2^hammingweight(3*n); \\ Joerg Arndt, Mar 10 2015

Crossrefs

Cf. A000120, A001316, A008287, A036555, A071053.

Sequence in context: A369757 A365489 A202686 * A132383 A115639 A242954

Adjacent sequences: A134657 A134658 A134659 * A134661 A134662 A134663

Keyword

nonn

Author

Steven Finch, Jan 25 2008

Status

approved