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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 2, 12, 12, 3, 108, 120, 24, 4, 1280, 1520, 280, 40, 5, 18750, 23400, 3930, 510, 60, 6, 326592, 423360, 65016, 7644, 840, 84, 7, 6588344, 8800008, 1241464, 132552, 13440, 1288, 112, 8, 150994944, 206622720, 26911296, 2622528, 244944, 22032, 1872, 144, 9
Offset: 1

Views

Author

Walt Rorie-Baety, Aug 15 2013

Keywords

Examples

			T(1,1) =  1: [1].
T(2,1) =  2: [1,2], [2,1].
T(2,2) =  2: [1,1], [2,2].
T(3,1) = 12: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3].
T(3,2) = 12: [1,1,2], [1,1,3], [1,2,2], [1,3,3], [2,1,1], [2,2,1], [2,2,3], [2,3,3], [3,1,1], [3,2,2], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
.       1;
.       2,       2;
.      12,      12,       3;
.     108,     120,      24,      4;
.    1280,    1520,     280,     40,     5;
.   18750,   23400,    3930,    510,    60,    6;
.  326592,  423360,   65016,   7644,   840,   84,   7;
. 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;

Links

Crossrefs

Row sums give: A000312.
Column k=1 gives: A055897.
Main diagonal gives: A000027.
Lower diagonal gives: 2*A180291.
Cf. A228194, A228273, A228617.

Programs

  • Maple
    T:= proc(n) option remember; local b; b:=
      proc(m, s, i) option remember; `if`(m>i or s>m, 0,
        `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
         b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
      end; forget(b);
      seq(add(b(k, s, n), s=1..k), k=1..n)
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 18 2013
  • Mathematica
    T[n_] := T[n] = Module[{b}, b[m_, s_, i_] := b[m, s, i] = If[m>i || s>m, 0, If[i == 1, n, If[s == 1, (n-1)*Sum[b[m, h, i-1], {h, 1, m}], b[m, s-1, i-1] + If[s == m, b[m-1, s-1, i-1], 0]]]]; Table[Sum[b[k, s, n], {s, 1, k}], {k, 1, n}]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A228194(n). - Alois P. Heinz, Dec 23 2020

A228250 Total sum A(n,k) of lengths of longest contiguous subsequences with the same value over all s in {1,...,n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 16, 12, 4, 0, 0, 5, 38, 45, 20, 5, 0, 0, 6, 86, 156, 96, 30, 6, 0, 0, 7, 188, 519, 436, 175, 42, 7, 0, 0, 8, 404, 1680, 1916, 980, 288, 56, 8, 0, 0, 9, 856, 5349, 8232, 5345, 1914, 441, 72, 9, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 18 2013

Keywords

Examples

			A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 16 = 3+2+1+2+2+1+2+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].
Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 1,  2,   3,    4,     5,      6,       7, ...
  0, 2,  6,  16,   38,    86,    188,     404, ...
  0, 3, 12,  45,  156,   519,   1680,    5349, ...
  0, 4, 20,  96,  436,  1916,   8232,   34840, ...
  0, 5, 30, 175,  980,  5345,  28610,  151115, ...
  0, 6, 42, 288, 1914, 12450,  79716,  504492, ...
  0, 7, 56, 441, 3388, 25571, 190428, 1403689, ...

Links

Crossrefs

Columns k=0-3 give: A000004, A001477, A002378, A152618(n+1).
Rows n=0-2 give: A000004, A001477, 2*A102712.
Main diagonal gives: A228194.
Cf. A228275.

Programs

  • Maple
    b:= proc(n, m, s, i) option remember; `if`(m>i or s>m, 0,
      `if`(i=0, 1, `if`(i=1, n, `if`(s=1, (n-1)*add(
         b(n, m, h, i-1), h=1..m), b(n, m, s-1, i-1)+
      `if`(s=m, b(n, m-1, s-1, i-1), 0)))))
    end:
A:= (n, k)-> add(m*add(b(n, m, s, k), s=1..m), m=1..k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    b[n_, m_, s_, i_] := b[n, m, s, i] = If[m>i || s>m, 0, If[i == 0, 1, If[i == 1, n, If[s == 1, (n-1)*Sum[b[n, m, h, i-1], {h, 1, m}], b[n, m, s-1, i-1] + If[s == m, b[n, m-1, s-1, i-1], 0]]]]]; A[n_, k_] := Sum[m*Sum[b[n, m, s, k], {s, 1, m}], {m, 1, k}]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A228618 Sum of lengths of shortest runs with the same value over all s in {1,...,n}^n.

Original entry on oeis.org

0, 1, 6, 33, 280, 3185, 46956, 824593, 16782872, 387446193, 10000154620, 285312472621, 8916105651300, 302875136889793, 11112007033351602, 437893891702463085, 18446744083262649616, 827240261951698484129, 39346408075791905146044, 1978419655663922695962061
Offset: 0

Views

Author

Alois P. Heinz, Aug 27 2013

Keywords

Examples

			a(3) = 33 = 3 + 12*1 + 3 + 12*1 + 3: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [1,3,3], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,2,2], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2], [3,3,3].

Links

Crossrefs

Cf. A228194.

Formula

a(n) = Sum_{k=0..n} k*A228617(n,k).
a(n) ~ n^n. - Vaclav Kotesovec, Sep 06 2014
Showing 1-3 of 3 results.

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