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%I #43 Oct 28 2022 08:56:05
%S 0,1,16,17,256,257,272,273,4096,4097,4112,4113,4352,4353,4368,4369,
%T 65536,65537,65552,65553,65792,65793,65808,65809,69632,69633,69648,
%U 69649,69888,69889,69904,69905,1048576,1048577,1048592,1048593,1048832
%N a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.
%C Numbers whose set of base 16 digits is {0,1}.
%C a(n) = Xpower(n,4). - _Antti Karttunen_, Apr 26 1999
%C Sums of distinct powers of 16.
%C 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
%H Vincenzo Librandi, <a href="/A033052/b033052.txt">Table of n, a(n) for n = 0..1000</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)*16^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
%F a(n) = A097262(n)/15.
%F a(2n) = 16*a(n), a(2n+1) = a(2n)+1.
%F a(n) = Sum_{k>=0} A030308(n,k)*16^k. - _Philippe Deléham_, Oct 19 2011
%F G.f.: (1/(1 - x))*Sum_{k>=0} 16^k*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jun 04 2017
%t FromDigits[#,16]&/@Tuples[{0,1},5] (* _Vincenzo Librandi_, Jun 04 2012 *)
%o (Magma) [n: n in [1..1050000] | Set(IntegerToSequence(n, 16)) subset {0, 1}]; // _Vincenzo Librandi_, May 04 2012
%o (PARI) a(n)=n=Vecrev(binary(n));sum(i=1,#n,n[i]<<(4*i))>>4 \\ _Charles R Greathouse IV_, Sep 23 2012
%o (C) int a_next(int a_n) { return (a_n + 0xeeeeeeef) & 0x11111111; } /* _Falk Hüffner_, Jan 24 2022 */
%Y Cf. A000695, A005836, A033042-A033051.
%Y Column 4 of A048723. Row 15 of array A104257.
%K nonn,base,easy
%O 0,3
%A _Clark Kimberling_
%E Extended by _Ray Chandler_, Aug 03 2004
%E Simpler definition from _Ralf Stephan_, Jun 18 2005
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