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

Author

Roger L. Bagula, Mar 15 2006

Keywords

• nonn

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

Crossrefs

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

Programs

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}]

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

Extensions

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