



0, 1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40, 42, 64, 65, 68, 69, 80, 81, 84, 85, 128, 130, 136, 138, 160, 162, 168, 170, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 512, 514, 520, 522, 544, 546, 552, 554, 640, 642, 648, 650
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OFFSET

1,3


COMMENTS

Essentially the same as A032937: 0 followed by terms of A032937.  R. J. Mathar, Jun 15 2008
Previous name was: If A = {a_1, a_2, a_3...} is the Moserde Bruijn sequence A000695 (consisting of sums of distinct powers of 4) and A' = {2a_1, 2a_2, 2a_3...} then this sequence, let's call it B, is the union of A and A'. Its significance, alluded to in the entry for the Moserde Bruijn sequence, is that its sumset, B+B, = {b_i + b_j : i, j natural numbers} consists of the nonnegative integers; and it is the fastestgrowing sequence with this property. It can also be described as a "basis of order two for the nonnegative integers".
The sequence is the fastest growing with this property in the sense that a(n) ~ n^2, and any sequence with this property is O(n^2).  Franklin T. AdamsWatters, Jul 27 2015
Or, base 2 representation Sum{d(i)*2^(mi): i=0,1,...,m} has even d(i) for all odd i.
Union of A000695 and 2*A000695.  Ralf Stephan, May 05 2004
Union of A000695 and A062880.  Franklin T. AdamsWatters, Aug 30 2014


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.


FORMULA

G.f.: sum(i>=1, T(i, x) + U(i, x) ), where
T := (k,x) > x^(2^k1)*V(k,x);
U := (k,x) > 2*x^(3*2^(k1)1)*V(k,x); and
V := (k,x) > (1x^(2^(k1)))*(4^(k1) + sum(4^j*x^(2^j)/(1+x^(2^j)), j = 0..k2))/(1x);
Generating function. Define V(k) := [4^(k1) + Sum ( j=0 to k2, 4^j * x^(2^j)/(1+x^(2^j)) )] * (1x^(2^(k1)))/(1x) and T(k) := (x^(2^k1) * V(k), U(k) := x^(3*2^(k1)1) * V(k) then G.f. is Sum ( i >= 1, T(i) + U(i) ). Functional equation: if the sequence is a(n), n = 1, 2, 3, ... and h(x) := Sum ( n >= 1, x^a(n) ) then h(x) satisfies the following functional equation: (1 + x^2)*h(x^4)  (1  x)*h(x^2)  x*h(x) + x^2 = 0.


EXAMPLE

All nonnegative integers can be represented in the form b_i + b_j; e.g. 6 = 5+1, 7 = 5+2, 8 = 0+8, 9 = 4+5


MATHEMATICA

nmax = 100;
b[n_] := FromDigits[IntegerDigits[n, 2], 4];
Union[A000695 = b /@ Range[0, nmax], 2 A000695][[1 ;; nmax+1]] (* JeanFrançois Alcover, Oct 28 2019 *)


PROG

(PARI) for(n=0, 350, b=binary(n):l=length(b); if(sum(i=1, floor(l/2), component(b, 2*i))==0, print1(n, ", ")))
(Haskell)
a126684 n = a126684_list !! (n1)
a126684_list = tail $ m a000695_list $ map (* 2) a000695_list where
m xs'@(x:xs) ys'@(y:ys)  x < y = x : m xs ys'
 otherwise = y : m xs' ys
 Reinhard Zumkeller, Dec 03 2011


CROSSREFS

Cf. A000695, A062880, A033053, A032937.
Sequence in context: A105425 A199799 A032937 * A089653 A180252 A191203
Adjacent sequences: A126681 A126682 A126683 * A126685 A126686 A126687


KEYWORD

easy,nonn


AUTHOR

Jonathan Deane, Feb 15 2007, May 04 2007


EXTENSIONS

New name (using comment from Ralf Stephan) from Joerg Arndt, Aug 31 2014


STATUS

approved



