A211164 Number of compositions of n with at most one odd part.

1, 1, 1, 3, 2, 8, 4, 20, 8, 48, 16, 112, 32, 256, 64, 576, 128, 1280, 256, 2816, 512, 6144, 1024, 13312, 2048, 28672, 4096, 61440, 8192, 131072, 16384, 278528, 32768, 589824, 65536, 1245184, 131072, 2621440, 262144, 5505024, 524288, 11534336, 1048576, 24117248

Offset: 0

Author

Alois P. Heinz, Jan 30 2013

Keywords

• nonn
• easy

Examples

			a(3) = 3: [3], [1,2], [2,1].
a(4) = 2: [4], [2,2].
a(5) = 8: [5], [3,2], [2,3], [1,4], [4,1], [1,2,2], [2,1,2], [2,2,1].
a(6) = 4: [6], [4,2], [2,4], [2,2,2].
a(8) = 8: [8], [4,4], [2,6], [6,2], [2,2,4], [4,2,2], [2,4,2], [2,2,2,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4)

Crossrefs

Bisection gives: A011782 (even part), A001792 (odd part).

Cf. A208354.

Programs

Maple

a:= n-> `if`(n<2, 1, 2^iquo(n-2, 2) *
        `if`(irem(n, 2)=0, 1, iquo(n+3, 2))):
seq(a(n), n=0..60);

PARI

Vec((1-x)^2*(1+x)*(1+2*x)/(1-2*x^2)^2 + O(x^50)) \\ Colin Barker, May 07 2016

Formula

G.f.: -(2*x^4-x^3-3*x^2+x+1)/(-4*x^4+4*x^2-1).

From Colin Barker, May 07 2016: (Start)

a(n) = 2^((n-7)/2+5/2) for n>0 and even.

a(n) = 2^((n-7)/2)*(2*n+6) for n>0 and odd.

a(n) = 4*a(n-2)-4*a(n-4) for n>4.

(End)