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A244714 Number of compositions of n with exactly 2 transitions between different parts.

%I A244714 #12 Aug 29 2021 04:32:58
%S A244714 1,4,14,29,56,100,148,230,322,446,573,778,953,1215,1456,1806,2134,
%T A244714 2542,2944,3477,3968,4600,5186,5872,6657,7446,8304,9217,10258,11172,
%U A244714 12465,13564,14867,16072,17716,18816,20832,22055,24144,25504,27904,29168,32051,33375
%N A244714 Number of compositions of n with exactly 2 transitions between different parts.
%H A244714 Alois P. Heinz, <a href="/A244714/b244714.txt">Table of n, a(n) for n = 4..1000</a>
%e A244714 a(4) = 1: [1,2,1].
%e A244714 a(5) = 4: [1,1,2,1], [1,2,1,1], [1,3,1], [2,1,2].
%p A244714 b:= proc(n, v) option remember; `if`(n=0, [1, 0$2],
%p A244714       add(`if`(v in [0, i], b(n-i, `if`(i<=n-i, i, -1)),
%p A244714       [0, b(n-i, `if`(i<=n-i, i, -1))[1..2][]]), i=1..n))
%p A244714     end:
%p A244714 a:= n-> b(n, 0)[3]:
%p A244714 seq(a(n), n=4..60);
%t A244714 b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n - i, i]*
%t A244714      If[v == 0 || v == i, 1, x], {i, n}]]];
%t A244714 a[n_] := Coefficient[b[n, 0], x, 2];
%t A244714 Table[a[n], {n, 4, 60}] (* _Jean-François Alcover_, Aug 29 2021, after A238279 Maple code *)
%Y A244714 Column k=2 of A238279.
%K A244714 nonn
%O A244714 4,2
%A A244714 _Joerg Arndt_ and _Alois P. Heinz_, Jul 04 2014