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.

1, 0, 3, 6, 10, 23, 50, 99, 200, 404, 805, 1599, 3166, 6225, 12223, 23934, 46713, 90995, 176935, 343395, 665474, 1287918, 2489467, 4806805, 9272272, 17870317, 34414163, 66226890, 127365537, 244803475, 470278815, 902997083, 1733124564, 3325087228, 6377076320

Offset: 2

Author

Christian G. Bower, Jan 23 2004

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=2 of A091613.

Cf. A128695.

Programs

Maple

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 2) -b(n, 0, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Feb 08 2017

Mathematica

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}]];
a[n_] := b[n, 0, 2] - b[n, 0, 1];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)
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 *)

Formula

a(n) ~ c * d^n, where d = 1.9107639262818041675000243699745706859... (see A128695), c = 0.499300813712837808621944870186032611... - Vaclav Kotesovec, Sep 21 2019

a(n) = A128695(n) - A003242(n). - Vaclav Kotesovec, Jul 07 2020