%I #12 Feb 08 2017 08:49:53

%S 2,4,8,24,60,152,400,1032,2656,6876,17776,45912,118664,306680,792480,

%T 2047984,5292564,13677160,35345112,91340568,236046088,610000528,

%U 1576390448,4073776744,10527631456,27205966108,70306845872,181690021616,469531293752,1213383282936

%N Number of compositions of n such that no two adjacent parts are equal, allowing 0.

%H Alois P. Heinz, <a href="/A114900/b114900.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Knopfmacher and H. Prodinger, <a href="http://www.sciencedirect.com/science/article/pii/S0195669898902165">On Carlitz compositions</a>, European Journal of Combinatorics, Vol. 19, No. 5, Jul 1998, pp. 579-589.

%F G.f.: 2*B(x)/(2-B(x)) where B(x) is g.f. of A003242.

%e The 8 compositions of 2 are 2, 2+0, 1+0+1, 1+0+1+0, 0+2, 0+2+0, 0+1+0+1, 0+1+0+1+0.

%p b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, 1, 2),

%p add(`if`(i=j, 0, b(n-j, `if`(j>n-j, -1, j))), j=0..n))

%p end:

%p a:= n-> b(n, -1):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 04 2015

%t b[n_, i_] := b[n, i] = If[n==0, If[i==0, 1, 2], Sum[If[i==j, 0, b[n-j, If[j > n-j, -1, j]]], {j, 0, n}]]; a[n_] := b[n, -1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 08 2017, after _Alois P. Heinz_ *)

%Y Cf. A003242, A114903.

%K nonn

%O 0,1

%A _Christian G. Bower_, Jan 05 2006

%E Replaced broken link, _Vaclav Kotesovec_, May 01 2014