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A091616 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 2 times and no part is repeated consecutively more than 2 times.

%I A091616 #19 Jul 07 2020 09:51:04
%S A091616 1,0,3,6,10,23,50,99,200,404,805,1599,3166,6225,12223,23934,46713,
%T A091616 90995,176935,343395,665474,1287918,2489467,4806805,9272272,17870317,
%U A091616 34414163,66226890,127365537,244803475,470278815,902997083,1733124564,3325087228,6377076320
%N A091616 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 2 times and no part is repeated consecutively more than 2 times.
%H A091616 Alois P. Heinz, <a href="/A091616/b091616.txt">Table of n, a(n) for n = 2..1000</a>
%F A091616 a(n) ~ c * d^n, where d = 1.9107639262818041675000243699745706859... (see A128695), c = 0.499300813712837808621944870186032611... - _Vaclav Kotesovec_, Sep 21 2019
%F A091616 a(n) = A128695(n) - A003242(n). - _Vaclav Kotesovec_, Jul 07 2020
%p A091616 b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
%p A091616       i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
%p A091616     end:
%p A091616 a:= n-> b(n, 0, 2) -b(n, 0, 1):
%p A091616 seq(a(n), n=2..50);  # _Alois P. Heinz_, Feb 08 2017
%t A091616 b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
%t A091616 a[n_] := b[n, 0, 2] - b[n, 0, 1];
%t A091616 Table[a[n], {n, 2, 50}] (* _Jean-François Alcover_, May 21 2018, after _Alois P. Heinz_ *)
%t A091616 nmax = 50; Drop[CoefficientList[Series[1/(1 - Sum[(x^k + x^(2*k))/(1 + x^k + x^(2*k)), {k, 1, nmax}]) - 1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x], 2] (* _Vaclav Kotesovec_, Jul 07 2020 *)
%Y A091616 Column k=2 of A091613.
%Y A091616 Cf. A128695.
%K A091616 nonn
%O A091616 2,3
%A A091616 _Christian G. Bower_, Jan 23 2004