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%I #6 Mar 22 2018 17:57:10
%S 1,1,0,1,2,1,1,3,3,3,7,9,6,10,20,20,20,36,50,54,75,109,126,156,233,
%T 302,352,480,676,838,1053,1447,1896,2374,3152,4225,5368,6923,9297,
%U 12133,15472,20353,26959,34779,45092,59551,77717,100475,131714,172949,224316,291987,383418
%N Number of compositions (ordered partitions) of n into triangular parts (A000217) such that no two adjacent parts are equal (Carlitz compositions).
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 + x^(k*(k+1)/2))).
%e a(12) = 6 because we have [3, 6, 3], [3, 1, 3, 1, 3, 1], [1, 10, 1], [1, 6, 1, 3, 1], [1, 3, 1, 6, 1] and [1, 3, 1, 3, 1, 3].
%t nmax = 52; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y Cf. A000217, A003242, A023361, A301503.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Mar 22 2018