login
A110316
a(n) is the number of different shapes of balanced binary trees with n nodes. The tree is balanced if the total number of nodes in the left and right branch of every node differ by at most one.
8
1, 1, 2, 1, 4, 4, 4, 1, 8, 16, 32, 16, 32, 16, 8, 1, 16, 64, 256, 256, 1024, 1024, 1024, 256, 1024, 1024, 1024, 256, 256, 64, 16, 1, 32, 256, 2048, 4096, 32768, 65536, 131072, 65536, 524288, 1048576, 2097152, 1048576, 2097152, 1048576, 524288, 65536, 524288
OFFSET
0,3
COMMENTS
The value of a(n) is always a power of 2.
REFERENCES
Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585
LINKS
S. Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv preprint arXiv:1107.3472 [math.CO], 2011.
Wikipedia, Blancmange curve
FORMULA
a(0) = a(1) = 1; a(2*n) = 2*a(n)*a(n-1); a(2*n+1) = a(n)*a(n).
a(n) = 1 <=> n in { A000225 }. - Orson R. L. Peters, Mar 12 2024
MAPLE
a:= proc(n) option remember; local r; `if`(n<2, 1,
`if`(irem(n, 2, 'r')=0, 2*a(r)*a(r-1), a(r)^2))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Apr 10 2013
MATHEMATICA
a[0] = a[1] = 1; a[n_] := a[n] = If[EvenQ[n], 2 a[n/2] a[n/2-1], a[(n-1)/2 ]^2]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 31 2016 *)
PROG
(Python)
def A110316(n): return 1<<(k:=n+1)-(sum(i.bit_count() for i in range(1, k))<<1)+k*(m:=k.bit_length())-(1<<m) # Chai Wah Wu, Mar 02 2023
CROSSREFS
Column k=2 of A221857. - Alois P. Heinz, Apr 17 2013
Cf. A000225.
Sequence in context: A140946 A008741 A369999 * A111975 A117250 A345674
KEYWORD
easy,nonn,look
AUTHOR
Jeffrey Barnett, Jun 23 2007
STATUS
approved