A261984 -id:A261984 - cp's OEIS frontend

A241902 Decimal expansion of a constant related to Carlitz compositions (A003242).

1, 7, 5, 0, 2, 4, 1, 2, 9, 1, 7, 1, 8, 3, 0, 9, 0, 3, 1, 2, 4, 9, 7, 3, 8, 6, 2, 4, 6, 3, 9, 8, 1, 5, 8, 7, 8, 7, 7, 8, 2, 0, 5, 8, 1, 8, 1, 3, 8, 1, 5, 9, 0, 5, 6, 1, 3, 1, 6, 5, 8, 6, 1, 3, 1, 7, 5, 1, 9, 3, 5, 1, 6, 7, 1, 5, 2, 0, 6, 0, 5, 0, 7, 7, 7, 4, 3, 8, 8, 7, 5, 6, 5, 7, 0, 9, 2, 4, 7, 1, 4, 1, 0, 0, 1

Offset: 1

Vaclav Kotesovec, May 01 2014

			1.7502412917183090312497386246398158787782...

Vaclav Kotesovec, Table of n, a(n) for n = 1..1500
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, p. 201.
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Cf. A003242, A224958, A212322, A106357, A261984, A106369.

RealDigits[r /. FindRoot[Exp[QPolyGamma[0, 1 + Pi*I/Log[r], r]] == r^(3/2)/(1-r), {r, 3/2}, WorkingPrecision -> 120], 10, 110][[1]] (* Vaclav Kotesovec, Jun 19 2023 *)

Equals lim n -> infinity A003242(n)^(1/n).

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.

Original entry on oeis.org

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

Alois P. Heinz, Sep 07 2015

			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;

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

Columns k=1-2 give: A261983, A261984.
Row sums give A261982.
Cf. A000142, A261960, A262191.

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);

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 *)

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

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

cp's OEIS Frontend

A241902 Decimal expansion of a constant related to Carlitz compositions (A003242).

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