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%I #20 Aug 22 2014 15:28:48
%S 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 1310,1867,2604,3648,5184,7346,10318,14506,20516,29022,40880,57548,
%U 81260,114810,161864,228092,321892,454444,640954,903715,1274998,1799320,2538218,3579714,5049954,7125359,10051844
%N Carlitz compositions of n into odd parts.
%C Carlitz compositions are compositions where adjacent parts are distinct (see A003242).
%H 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 G.f.: 1/( 1 - Sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) ). - _Geoffrey Critzer_, Nov 21 2013
%F 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 There are a(12) = 22 such compositions of 12:
%e [ 1] 1 3 1 3 1 3
%e [ 2] 1 3 1 7
%e [ 3] 1 3 5 3
%e [ 4] 1 3 7 1
%e [ 5] 1 5 1 5
%e [ 6] 1 7 1 3
%e [ 7] 1 7 3 1
%e [ 8] 1 11
%e [ 9] 3 1 3 1 3 1
%e [10] 3 1 3 5
%e [11] 3 1 5 3
%e [12] 3 1 7 1
%e [13] 3 5 1 3
%e [14] 3 5 3 1
%e [15] 3 9
%e [16] 5 1 5 1
%e [17] 5 3 1 3
%e [18] 5 7
%e [19] 7 1 3 1
%e [20] 7 5
%e [21] 9 3
%e [22] 11 1
%p b:= proc(n, t) option remember; `if`(n=0, 1,
%p add(`if`(j=t or irem(j, 2)=0, 0, b(n-j, j)), j=1..n))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Nov 08 2012
%t 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 Cf. A003242 (Carlitz compositions), A032021 (compositions into distinct odd parts), A032020 (compositions into distinct parts).
%K nonn
%O 0,5
%A _Joerg Arndt_, Nov 04 2012
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