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A298732 Number of compositions (ordered partitions) of n into parts > 1 such that no two adjacent parts are equal (Carlitz compositions).

%I A298732 #7 Jan 25 2018 19:22:29
%S A298732 1,0,1,1,1,3,3,6,7,14,18,30,45,66,107,157,245,369,569,862,1325,2020,
%T A298732 3078,4717,7183,10991,16769,25626,39117,59763,91264,139362,212893,
%U A298732 325060,496525,758258,1158079,1768634,2701162,4125320,6300303,9622247,14695253,22443451,34276405,52348435
%N A298732 Number of compositions (ordered partitions) of n into parts > 1 such that no two adjacent parts are equal (Carlitz compositions).
%H A298732 Alois P. Heinz, <a href="/A298732/b298732.txt">Table of n, a(n) for n = 0..1000</a>
%H A298732 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A298732 G.f.: 1/(1 - Sum_{k>=2} x^k/(1 + x^k)).
%e A298732 a(7) = 6 because we have [7], [5, 2], [4, 3], [3, 4], [2, 5] and [2, 3, 2].
%p A298732 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A298732       add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=2..n))
%p A298732     end:
%p A298732 a:= n-> b(n, 0):
%p A298732 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 25 2018
%t A298732 nmax = 45; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
%Y A298732 Cf. A000045, A002865, A003242, A011782, A025147, A032020, A114900, A155822, A212804, A218694.
%K A298732 nonn
%O A298732 0,6
%A A298732 _Ilya Gutkovskiy_, Jan 25 2018