A033044 Sums of distinct powers of 7.

0, 1, 7, 8, 49, 50, 56, 57, 343, 344, 350, 351, 392, 393, 399, 400, 2401, 2402, 2408, 2409, 2450, 2451, 2457, 2458, 2744, 2745, 2751, 2752, 2793, 2794, 2800, 2801, 16807, 16808, 16814, 16815, 16856, 16857, 16863, 16864, 17150, 17151, 17157

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Numbers without any base-7 digits greater than 1.

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

Note that this sequence has offset 1, in contrast to A000695 and all others among A033043-A033052. - M. F. Hasler, Feb 01 2016

Links

• T. D. Noe, Table of n, a(n) for n = 1..1024
• 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, p. 45.

Crossrefs

Cf. A000695, A005836, A033043-A033052, A037412.

Row 7 of array A104257.

Programs

Mathematica

t = Table[FromDigits[RealDigits[n, 2], 7], {n, 0, 100}]
(* Clark Kimberling, Aug 03 2012 *)
FromDigits[#,7]&/@Tuples[{0,1},6] (* Harvey P. Dale, Apr 30 2015 *)

PARI

A033044(n,b=7)=subst(Pol(binary(n-1)),'x,b) \\ M. F. Hasler, Feb 01 2016

PARI

a(n)=fromdigits(binary(n), 7) \\ Charles R Greathouse IV, Jan 11 2017

Formula

a(n) = Sum_{i=0..m} d(i)*7^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n) = A097253(n)/6.

a(2n) = 7*a(n), a(2n+1) = a(2n)+1.

a(n+1) = Sum_{k>=0} A030308(n,k)*7^k. - Philippe Deléham, Oct 17 2011

G.f.: (x/(1 - x))*Sum_{k>=0} 7^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extensions

Extended by Ray Chandler, Aug 03 2004

Karol Bacik has pointed out that the first three formulas do not match the sequence. - N. J. A. Sloane, Oct 20 2012