A211164 - OEIS

%I #27 May 07 2016 02:53:49

%S 1,1,1,3,2,8,4,20,8,48,16,112,32,256,64,576,128,1280,256,2816,512,

%T 6144,1024,13312,2048,28672,4096,61440,8192,131072,16384,278528,32768,

%U 589824,65536,1245184,131072,2621440,262144,5505024,524288,11534336,1048576,24117248

%N Number of compositions of n with at most one odd part.

%H Alois P. Heinz, <a href="/A211164/b211164.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4)

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

%F From _Colin Barker_, May 07 2016: (Start)

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

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

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

%F (End)

%e a(3) = 3: [3], [1,2], [2,1].

%e a(4) = 2: [4], [2,2].

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

%e a(6) = 4: [6], [4,2], [2,4], [2,2,2].

%e a(8) = 8: [8], [4,4], [2,6], [6,2], [2,2,4], [4,2,2], [2,4,2], [2,2,2,2].

%p a:= n-> `if`(n<2, 1, 2^iquo(n-2, 2) *

%p         `if`(irem(n, 2)=0, 1, iquo(n+3, 2))):

%p seq(a(n), n=0..60);

%o (PARI) Vec((1-x)^2*(1+x)*(1+2*x)/(1-2*x^2)^2 + O(x^50)) \\ _Colin Barker_, May 07 2016

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

%Y Cf. A208354.

%K nonn,easy

%O 0,4

%A _Alois P. Heinz_, Jan 30 2013