A106357 Number of compositions of n with exactly 1 adjacent equal pair of parts.

1, 0, 3, 6, 7, 20, 42, 72, 141, 280, 516, 976, 1853, 3420, 6361, 11844, 21819, 40192, 73942, 135452, 247828, 452776, 825252, 1501998, 2730159, 4954890, 8981360, 16261568, 29408708, 53130154, 95894384, 172917788, 311538169, 560831286

Offset: 2

Author

Christian G. Bower, Apr 29 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000
• A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Crossrefs

Column 1 of A106356. Cf. A003242.

Cf. A241902.

Programs

Maple

b:= proc(n, v) option remember; `if`(n=0, [1, 0],
      add((p-> `if`(i=v, [0, p[1]], p))(b(n-i, i)), i=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=2..45);  # Alois P. Heinz, May 09 2014

Mathematica

b[n_, v_] := b[n, v] = If[n == 0, {1, 0}, Sum[Function[p, If[i == v, {0, p[[1]]}, p]][b[n - i, i]], {i, 1, n}]];
a[n_] := b[n, 0][[2]];
a /@ Range[2, 45] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

PARI

C_x(N)={my(g=1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(C_x(N)*x^(2*i))/(1+x^i)^2 )/(1-sum(i=1,N,(x^i)/(1+x^i)))); Vec(h)}
A_x(40) \\ John Tyler Rascoe, May 16 2024

Formula

a(n) ~ c * d^n * n, where d = A241902 = 1.750241291718309031249738624639..., c = 0.04826600476992825168367... . - Vaclav Kotesovec, Sep 05 2014

G.f.: (Sum_{i>0} C(x)*x^(2*i)/(1+x^i)^2)/(1 - Sum_{i>0} x^i/(1+x^i)) where C(x) is the g.f. for A003242. - John Tyler Rascoe, May 16 2024

Extensions

Replaced broken link, Vaclav Kotesovec, May 01 2014