A261984 Number of compositions of n such that the minimal distance between two identical parts equals two.

0, 0, 0, 0, 1, 2, 3, 8, 16, 34, 57, 113, 213, 396, 733, 1333, 2419, 4400, 7934, 14321, 25687, 45947, 82085, 146410, 260547, 463021, 821669, 1456296, 2578051, 4559972, 8057373, 14225124, 25096606, 44246087, 77958821, 137283534, 241626535, 425079358, 747501363

Offset: 0

Author

Alois P. Heinz, Sep 07 2015

Keywords

• nonn

Examples

			a(4) = 1: 121.
a(5) = 2: 131, 212.
a(6) = 3: 141, 1212, 2121.
a(7) = 8: 151, 232, 313, 1213, 1312, 2131, 3121, 12121.
a(8) = 16: 161, 242, 323, 1214, 1232, 1313, 1412, 2123, 2141, 2321, 3131, 3212, 4121, 12131, 13121, 21212.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Column k=2 of A261981.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, 1, add(
     `if`(i=j, 0, g(n-j, j)), j=1..n))
    end:
b:= proc(n, i, m) option remember; `if`(n=0, 0, add(
     `if`(i=j, 0, `if`(j=m, g(n-j, j), b(n-j, j, i))), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..45);

Mathematica

g[n_, i_] := g[n, i] = If[n==0, 1, Sum[If[i==j, 0, g[n-j, j]], {j, 1, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n==0, 0, Sum[If[i==j, 0, If[j==m, g[n-j, j], b[n-j, j, i]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 45] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) ~ A003242(n). - Vaclav Kotesovec, Sep 08 2015