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

2,3

Links

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

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.

Example

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;

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 *)

Crossrefs

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

Row sums give A261982.

Cf. A000142, A261960, A262191.

Sequence in context: A304526 A333352 A260253 * A153265 A331153 A331149

Adjacent sequences: A261978 A261979 A261980 * A261982 A261983 A261984

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 07 2015

Status

approved