A033048 Sums of distinct powers of 12.

0, 1, 12, 13, 144, 145, 156, 157, 1728, 1729, 1740, 1741, 1872, 1873, 1884, 1885, 20736, 20737, 20748, 20749, 20880, 20881, 20892, 20893, 22464, 22465, 22476, 22477, 22608, 22609, 22620, 22621, 248832, 248833, 248844, 248845, 248976

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Numbers without any base-12 digits greater than 1.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1023
• 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.
• Eric Weisstein's World of Mathematics, Duodecimal
• Wikipedia, Duodecimal

Crossrefs

Subsequence of A102487.

Cf. A000695, A005836, A033042-A033052.

Row 11 of array A104257.

Programs

Haskell

import Data.List (unfoldr)
a033048 n = a033048_list !! (n-1)
a033048_list = filter (all (< 2) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [1..]
-- Reinhard Zumkeller, Apr 17 2011

Mathematica

With[{k = 12}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)

PARI

{maxn=37;
for(vv=0,maxn,
bvv=binary(vv);
ll=length(bvv);texp=0;btod=0;
forstep(i=ll,1,-1,btod=btod+bvv[i]*12^texp;texp++);
print1(btod,", "))}
\\ Douglas Latimer, Apr 16 2012

PARI

a(n)=fromdigits(binary(n),12) \\ Charles R Greathouse IV, Jan 11 2017

Formula

a(n) = Sum_{i=0..m} d(i)*12^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n) = A097258(n)/11.

a(2n) = 12*a(n), a(2n+1) = a(2n)+1.

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(k) = 12^k = A001021(k). - Philippe Deléham, Oct 19 2011

G.f.: (1/(1 - x))*Sum_{k>=0} 12^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extensions

Extended by Ray Chandler, Aug 03 2004