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A285994
Number of increasing runs in all Carlitz compositions of n.
2
0, 1, 1, 4, 6, 11, 26, 46, 84, 167, 313, 576, 1086, 2016, 3710, 6876, 12660, 23196, 42542, 77798, 141910, 258648, 470558, 854644, 1550588, 2809620, 5084588, 9192349, 16601714, 29953754, 53997062, 97257129, 175033355, 314771224, 565664138, 1015841191
OFFSET
0,4
COMMENTS
No two adjacent parts of a Carlitz composition are equal.
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/3)} (k+1) * A241701(n,k) for n>0, a(0) = 0.
EXAMPLE
a(1) = 1: (1).
a(2) = 1: (2).
a(3) = 4: (12), (2)(1), (3).
a(4) = 6: (12)(1), (13), (3)(1), (4).
a(5) = 11: (2)(12), (13)(1), (23), (3)(2), (14), (4)(1), (5).
MAPLE
b:= proc(n, l) option remember; `if`(n=0, [1, 0], add(`if`(j=l, 0,
(p-> p+`if`(j>l, [0, p[1]], 0))(b(n-j, j))), j=1..n))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..40);
MATHEMATICA
b[n_, l_] := b[n, l] = If[n == 0, {1, 0}, Sum[If[j == l, {0, 0}, Function[p, p + If[j > l, {0, p[[1]]}, 0]][b[n - j, j]]], {j, 1, n}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2022, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A219520 A336343 A330459 * A290651 A358913 A066155
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 30 2017
STATUS
approved