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

1,3

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.

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

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 *)

Prog

(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

Crossrefs

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

Row 7 of array A104257.

Sequence in context: A047192 A342611 A270009 * A025630 A036566 A116554

Adjacent sequences: A033041 A033042 A033043 * A033045 A033046 A033047

Keyword

nonn,base,easy

Author

Clark Kimberling

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

Status

approved