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.
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
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
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Project Euler, Problem 427: n-sequences
Crossrefs
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 *)