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

0,4

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)

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)

Example

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].

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);

Prog

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

Crossrefs

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

Cf. A208354.

Sequence in context: A162728 A127300 A129199 * A097018 A127541 A053219

Adjacent sequences: A211161 A211162 A211163 * A211165 A211166 A211167

Keyword

nonn,easy

Author

Alois P. Heinz, Jan 30 2013

Status

approved