OFFSET
1,2
COMMENTS
The maximum is always obtained by taking i as the power of 2 nearest to n/2. - Anna de Mier, Mar 12 2012
a(n) is the number of (binary) max-heaps on n-1 elements from the set {0,1}. a(7) = 16: 000000, 100000, 101000, 101001, 110000, 110010, 110100, 110110, 111000, 111001, 111010, 111011, 111100, 111101, 111110, 111111. - Alois P. Heinz, Jul 09 2019
REFERENCES
A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Graphs Combin., 28 (2012), 265-275.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..5652
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850. See Section 4.2.
A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Elect. Notes Discr. Math 34 (2009) 489-493
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
FORMULA
a(n) = 1 + max_{i=1..n-1} a(i)*a(n-i) for n > 1, a(1) = 1.
From Alois P. Heinz, Jul 09 2019: (Start)
a(n) = Sum_{k=0..n-1} A309049(n-1,k).
a(2^(n-1)) = A003095(n). (End)
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, (g-> (f->
1+b(f)*b(n-1-f))(min(g-1, n-g/2)))(2^ilog2(n)))
end:
a:= n-> b(n-1):
seq(a(n), n=1..50); # Alois P. Heinz, Jul 09 2019
MATHEMATICA
a[n_] := a[n] = 1 + Max[Table[a[i] a[n-i], {i, n-1}]]; a[1] = 1;
Array[a, 50] (* Jean-François Alcover, Apr 30 2020 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Franklin T. Adams-Watters, Mar 15 2004
STATUS
approved