A033048 - OEIS

%I #48 Oct 31 2022 02:09:36

%S 0,1,12,13,144,145,156,157,1728,1729,1740,1741,1872,1873,1884,1885,

%T 20736,20737,20748,20749,20880,20881,20892,20893,22464,22465,22476,

%U 22477,22608,22609,22620,22621,248832,248833,248844,248845,248976

%N Sums of distinct powers of 12.

%C Numbers without any base-12 digits greater than 1.

%H T. D. Noe, <a href="/A033048/b033048.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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Duodecimal.html">Duodecimal</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Duodecimal">Duodecimal</a>

%F 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.

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

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

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

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

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

%o (PARI) {maxn=37;

%o for(vv=0,maxn,

%o bvv=binary(vv);

%o ll=length(bvv);texp=0;btod=0;

%o forstep(i=ll,1,-1,btod=btod+bvv[i]*12^texp;texp++);

%o print1(btod,", "))}

%o \\ _Douglas Latimer_, Apr 16 2012

%o (PARI) a(n)=fromdigits(binary(n),12) \\ _Charles R Greathouse IV_, Jan 11 2017

%o (Haskell)

%o import Data.List (unfoldr)

%o a033048 n = a033048_list !! (n-1)

%o a033048_list = filter (all (< 2) . unfoldr (\x ->

%o    if x == 0 then Nothing else Just $ swap $ divMod x 12)) [1..]

%o -- _Reinhard Zumkeller_, Apr 17 2011

%Y Subsequence of A102487.

%Y Cf. A000695, A005836, A033042-A033052.

%Y Row 11 of array A104257.

%K nonn,base,easy

%O 0,3

%A _Clark Kimberling_

%E Extended by _Ray Chandler_, Aug 03 2004