OFFSET
0,3
COMMENTS
Numbers whose set of base 16 digits is {0,1}.
a(n) = Xpower(n,4). - Antti Karttunen, Apr 26 1999
Sums of distinct powers of 16.
For every nonnegative n, A000695(n) is a unique sum of the form a(k) + 4a(l). Thus every nonnegative n is a unique sum of the form a(p) + 2a(q) + 4a(r) + 8a(s). This gives a one-to-one map of the set N_0 of all nonnegative integers to (N_0)^4. Furthermore, if, for a fixed positive integer m, to consider all sums of distinct powers of 4^m, then one can obtain a one-to-one map of the set N_0 to (N_0)^(2^m). - Vladimir Shevelev, Nov 14 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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)*16^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097262(n)/15.
a(2n) = 16*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*16^k. - Philippe Deléham, Oct 19 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 16^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017
MATHEMATICA
FromDigits[#, 16]&/@Tuples[{0, 1}, 5] (* Vincenzo Librandi, Jun 04 2012 *)
PROG
(Magma) [n: n in [1..1050000] | Set(IntegerToSequence(n, 16)) subset {0, 1}]; // Vincenzo Librandi, May 04 2012
(PARI) a(n)=n=Vecrev(binary(n)); sum(i=1, #n, n[i]<<(4*i))>>4 \\ Charles R Greathouse IV, Sep 23 2012
(C) int a_next(int a_n) { return (a_n + 0xeeeeeeef) & 0x11111111; } /* Falk Hüffner, Jan 24 2022 */
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Aug 03 2004
Simpler definition from Ralf Stephan, Jun 18 2005
STATUS
approved