A106358 Number of compositions of n with exactly 2 adjacent equal parts (2 pairs or 1 triple.).

1, 0, 2, 8, 10, 22, 58, 112, 219, 466, 920, 1787, 3600, 7025, 13532, 26315, 50625, 96775, 185000, 351714, 665942, 1258649, 2371219, 4454004, 8348735, 15612146, 29128863, 54245790, 100828939, 187074952, 346527510, 640878692, 1183480187

Offset: 3

Author

Christian G. Bower, Apr 29 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000
• A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Crossrefs

Column 2 of A106356. Cf. A003242.

Cf. A241902.

Programs

Maple

b:= proc(n, v) option remember; `if`(n=0, [1, 0$2], add((
      p->`if`(i=v, [0, p[1..2][]], p))(b(n-i, i)), i=1..n))
    end:
a:= n-> b(n, 0)[3]:
seq(a(n), n=3..45);  # Alois P. Heinz, Jun 24 2014

Mathematica

b[n_, v_] := b[n, v] = If[n==0, {1, 0, 0}, Sum[If[i==v, Prepend[#[[1;;2]], 0], #]&[b[n-i, i]], {i, 1, n}]];
a[n_] := b[n, 0][[3]];
a /@ Range[3, 45] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n^2, where d = 1.7502412917183090312497386246... (see A241902), c = 0.0025523594118210599072896951... . - Vaclav Kotesovec, Aug 25 2014