A261981 Number T(n,k) of compositions of n such that k is the minimal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=floor((sqrt(8*n-7)-1)/2), read by rows.

1, 1, 4, 1, 9, 2, 18, 3, 41, 8, 2, 89, 16, 4, 185, 34, 10, 388, 57, 10, 810, 113, 30, 6, 1670, 213, 52, 12, 3435, 396, 104, 28, 7040, 733, 176, 50, 14360, 1333, 278, 62, 29226, 2419, 512, 152, 24, 59347, 4400, 878, 246, 48, 120229, 7934, 1492, 458, 108

Offset: 2

Author

Alois P. Heinz, Sep 07 2015

Keywords

• nonn
• tabf

Examples

			T(5,1) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
T(5,2) = 2: 131, 212.
T(7,2) = 8: 151, 313, 232, 3121, 1213, 2131, 1312, 12121.
T(7,3) = 2: 1231, 1321.
Triangle T(n,k) begins:
n\k:     1     2    3    4   5
---+---------------------------
02 :     1;
03 :     1;
04 :     4,    1;
05 :     9,    2;
06 :    18,    3;
07 :    41,    8,   2;
08 :    89,   16,   4;
09 :   185,   34,  10;
10 :   388,   57,  10;
11 :   810,  113,  30,   6;
12 :  1670,  213,  52,  12;
13 :  3435,  396, 104,  28;
14 :  7040,  733, 176,  50;
15 : 14360, 1333, 278,  62;
16 : 29226, 2419, 512, 152, 24;
		

Links

• Alois P. Heinz, Rows n = 2..55, flattened

Crossrefs

Columns k=1-2 give: A261983, A261984.

Row sums give A261982.

Cf. A000142, A261960, A262191.

Programs

Maple

b:= proc(n, l) option remember;
      `if`(n=0, 1, add(`if`(j in l, 0, b(n-j,
      `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
    end:
T:= (n, k)-> b(n, [0$(k-1)])-b(n, [0$k]):
seq(seq(T(n, k), k=1..floor((sqrt(8*n-7)-1)/2)), n=2..20);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]];
A[n_, k_] := b[n, Array[0&, Min[n, k]]];
T[n_, k_] := A[n, k-1] - A[n, k];
Table[T[n, k], {n, 2, 20}, {k, 1, Floor[(Sqrt[8*n-7]-1)/2]}] // Flatten (* Jean-François Alcover, Apr 13 2017, after Alois P. Heinz *)

Formula

T(n,k) = A261960(n,k-1) - A261960(n,k).

T((n+1)*(n+2)/2+1,n+1) = A000142(n) for n>=0.