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A218694 Carlitz compositions of n into odd parts.

%I A218694 #20 Aug 22 2014 15:28:48
%S A218694 1,1,0,1,2,2,2,3,6,9,10,13,22,32,40,56,86,122,164,229,332,474,656,914,
%T A218694 1310,1867,2604,3648,5184,7346,10318,14506,20516,29022,40880,57548,
%U A218694 81260,114810,161864,228092,321892,454444,640954,903715,1274998,1799320,2538218,3579714,5049954,7125359,10051844
%N A218694 Carlitz compositions of n into odd parts.
%C A218694 Carlitz compositions are compositions where adjacent parts are distinct (see A003242).
%H A218694 Joerg Arndt and Alois P. Heinz, <a href="/A218694/b218694.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..262 from Joerg Arndt)
%F A218694 G.f.: 1/( 1 - Sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) ). - _Geoffrey Critzer_, Nov 21 2013
%F A218694 a(n) ~ c / r^n, where r = 0.708865489663179258570259601255070249415... is the root of the equation sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) = 1, c = 0.3391570949344217123793275284038135702369824934927187... . - _Vaclav Kotesovec_, Aug 22 2014
%e A218694 There are a(12) = 22 such compositions of 12:
%e A218694 [ 1]  1 3 1 3 1 3
%e A218694 [ 2]  1 3 1 7
%e A218694 [ 3]  1 3 5 3
%e A218694 [ 4]  1 3 7 1
%e A218694 [ 5]  1 5 1 5
%e A218694 [ 6]  1 7 1 3
%e A218694 [ 7]  1 7 3 1
%e A218694 [ 8]  1 11
%e A218694 [ 9]  3 1 3 1 3 1
%e A218694 [10]  3 1 3 5
%e A218694 [11]  3 1 5 3
%e A218694 [12]  3 1 7 1
%e A218694 [13]  3 5 1 3
%e A218694 [14]  3 5 3 1
%e A218694 [15]  3 9
%e A218694 [16]  5 1 5 1
%e A218694 [17]  5 3 1 3
%e A218694 [18]  5 7
%e A218694 [19]  7 1 3 1
%e A218694 [20]  7 5
%e A218694 [21]  9 3
%e A218694 [22]  11 1
%p A218694 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A218694        add(`if`(j=t or irem(j, 2)=0, 0, b(n-j, j)), j=1..n))
%p A218694     end:
%p A218694 a:= n-> b(n, 0):
%p A218694 seq(a(n), n=0..70);  # _Alois P. Heinz_, Nov 08 2012
%t A218694 nn=20;CoefficientList[Series[1/(1-Sum[z^(2j+1)/(1+z^(2j+1)),{j,0,nn}]),{z,0,nn}],z] (* _Geoffrey Critzer_, Nov 21 2013 *)
%Y A218694 Cf. A003242 (Carlitz compositions), A032021 (compositions into distinct odd parts), A032020 (compositions into distinct parts).
%K A218694 nonn
%O A218694 0,5
%A A218694 _Joerg Arndt_, Nov 04 2012