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 A062880 Zero together with numbers which can be written as a sum of distinct odd powers of 2. 16

%I

%S 0,2,8,10,32,34,40,42,128,130,136,138,160,162,168,170,512,514,520,522,

%T 544,546,552,554,640,642,648,650,672,674,680,682,2048,2050,2056,2058,

%U 2080,2082,2088,2090,2176,2178,2184,2186,2208,2210,2216,2218,2560,2562

%N Zero together with numbers which can be written as a sum of distinct odd powers of 2.

%C Binary expansion of n does not contain 1-bits at even positions.

%C Integers whose base-4 representation consists of only 0s and 2s.

%C a(n)=2 A000695(n). Every nonnegative even number is a unique sum of the form a(k)+2a(l); moreover, this sequence is unique with such property. [_Vladimir Shevelev_, Nov 07 2008]

%C Also numbers such that the digital sum base 2 and the digital sum base 4 are in a ratio of 2:4. - _Michel Marcus_, Sep 23 2013

%H Reinhard Zumkeller, <a href="/A062880/b062880.txt">Table of n, a(n) for n = 0..10000</a>

%H D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance, <a href="http://dx.doi.org/10.5802/jtnb.457">On the binary expansions of algebraic numbers</a>, J. Théor. Nombres Bordeaux, 16 (2004), 487-518.

%H S. Eigen, A. Hajian, and S. Kalikow, <a href="http://dx.doi.org/10.1007/BF02787185">Ergodic transformations and sequences of integers</a>, Israel J. Math. 75 (1991), 119-128; Math. Rev. 1147294 (93c:28014).

%F From _Robert Israel_, Apr 10 2018: (Start)

%F a(2*n) = 4*a(n).

%F a(2*n+1) = 4*a(n)+2.

%F G.f. g(x) satisfies: g(x) = 4*(1+x)*g(x^2)+2*x/(1-x^2). (End)

%p [seq(a(j),j=0..100)]; a := n -> add((floor(n/(2^i)) mod 2)*(2^((2*i)+1)),i=0..floor_log_2(n+1));

%t b[n_] := BitAnd[n, Sum[2^k, {k, 0, Log[2, n] // Floor, 2}]]; Select[Range[ 0, 10^4], b[#] == 0&] (* _Jean-François Alcover_, Feb 28 2016 *)

%o a062880 n = a062880_list !! n

%o a062880_list = filter f [0..] where

%o f 0 = True

%o f x = (m == 0 || m == 2) && f x' where (x', m) = divMod x 4

%o -- _Reinhard Zumkeller_, Nov 20 2012

%Y Except for first term, n such that A063694(n) = 0. Binary expansion is given in A062033.

%Y Interpreted as Zeckendorf expansion: A062879. A062880[n] = 2*A000695[n]

%Y Central diagonal of arrays A163357 and A163359.

%K nonn,easy

%O 0,2

%A _Antti Karttunen_, Jun 26 2001

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Last modified November 17 10:34 EST 2018. Contains 317275 sequences. (Running on oeis4.)