OFFSET
0,3
COMMENTS
A 9-divide version of A084230.
The interest this one has is in the prime form of even odd 2^n+1, 2^n.
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
1, 0, 0, 0, 0, ...
9, 0, 0, 0, 0, ...
8, 1, 0, 0, 0, ...
0, 9, 0, 0, 0, ...
0, 8, 1, 0, 0, ...
0, 0, 9, 0, 0, ...
0, 0, 8, 1, 0, ...
...
Then M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
FORMULA
a(n) = Sum_{k=0..n-1} 8^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 8^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023
MAPLE
a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9*a(n/2) else 8*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..45);
MATHEMATICA
b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9*b[n/2]; b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Mar 15 2006
EXTENSIONS
Edited by N. J. A. Sloane, Apr 16 2006
STATUS
approved