A116522 a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,..., a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

0, 1, 7, 13, 49, 55, 91, 127, 343, 349, 385, 421, 637, 673, 889, 1105, 2401, 2407, 2443, 2479, 2695, 2731, 2947, 3163, 4459, 4495, 4711, 4927, 6223, 6439, 7735, 9031, 16807, 16813, 16849, 16885, 17101, 17137, 17353, 17569, 18865, 18901, 19117, 19333

Offset

0,3

Comments

A 7-divide version of A084230.

The Harborth: f(2^k) = 3^k suggests that a family of sequences of the form: f(2^k) = prime(n)^k.

From Gary W. Adamson, Aug 27 2016: (Start)

Let M = the production matrix below. Then lim_{k->infinity} M^k generates the sequence with offset 1 by extracting the left-shifted vector.

1, 0, 0, 0, 0, ...

7, 0, 0, 0, 0, ...

6, 1, 0, 0, 0, ...

0, 7, 0, 0, 0, ...

0, 6, 1, 0, 0, ...

0, 0, 7, 0, 0, ...

0, 0, 6, 1, 0, ...

...

The sequence divided by its aerated variant is (1, 7, 6, 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

G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + 7x + 6x^2).

a(n) = Sum_{k=0..n-1} 6^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

a(n) = Sum_{k=0..floor(log_2(n))} 6^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 7*a(n/2) else 6*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..47);
# second Maple program:
b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=6, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

Mathematica

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 7*b[n/2]; b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

Crossrefs

Cf. A000120, A006046, A077465, A130665, A116520, A130667, A161342, A360189.

Sequence in context: A267310 A153119 A166703 * A320462 A108056 A018562

Adjacent sequences: A116519 A116520 A116521 * A116523 A116524 A116525

Keyword

nonn

Author

Roger L. Bagula, Mar 15 2006

Extensions

Edited by N. J. A. Sloane, Apr 16 2005

Status

approved