A033046 - OEIS

%I #50 Aug 06 2024 21:22:21

%S 0,1,9,10,81,82,90,91,729,730,738,739,810,811,819,820,6561,6562,6570,

%T 6571,6642,6643,6651,6652,7290,7291,7299,7300,7371,7372,7380,7381,

%U 59049,59050,59058,59059,59130,59131,59139,59140,59778,59779,59787

%N Sums of distinct powers of 9.

%C Numbers without any base-9 digits greater than 1.

%C a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - _Philippe Deléham_, Oct 17 2011

%H T. D. Noe, <a href="/A033046/b033046.txt">Table of n, a(n) for n = 0..1023</a>

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 45.

%F a(n) = Sum_{i=0..m} d(i)*9^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

%F a(n) = A097255(n)/8.

%F a(2n) = 9*a(n), a(2n+1) = a(2n)+1.

%F a(n) = Sum_{k>=0} A030308(n,k)*9^k. - _Philippe Deléham_, Oct 17 2011

%F G.f.: (1/(1 - x))*Sum_{k>=0} 9^k*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jun 04 2017

%t FromDigits[#,9]&/@Tuples[{1,0},6]//Sort (* _Harvey P. Dale_, Sep 05 2017 *)

%o (PARI) A033046(n,b=9)=subst(Pol(binary(n)),'x,b) \\ _M. F. Hasler_, Feb 01 2016

%Y Cf. A000695, A005836, A033043-A033052.

%Y Row 9 of array A104257.

%K nonn,base,easy

%O 0,3

%A _Clark Kimberling_

%E Extended by _Ray Chandler_, Aug 03 2004