%I A126684
%S A126684 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,
%T A126684 136,138,160,162,168,170,256,257,260,261,272,273,276,277,320,321,324,
%U A126684 325,336,337,340,341,512,514,520,522,544,546,552,554,640,642,648,650
%N A126684 If A = {a_1, a_2, a_3...} is the Moser-de 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 Moser-de 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 fastest-growing sequence 
               with this property. It can also be described as a "basis of order 
               two for the nonnegative integers".
%C A126684 Essentially the same as A032937. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jun 15 2008
%F A126684 G.f. = sum(T(i, x) + U(i, x), i = 1..infinity), where
%F A126684 T := (k,x) -> x^(2^k-1)*V(k,x);
%F A126684 U := (k,x) -> 2*x^(3*2^(k-1)-1)*V(k,x); and
%F A126684 V := (k,x) -> (1-x^(2^(k-1)))*(4^(k-1) + sum(4^j*x^(2^j)/(1+x^(2^j)), 
               j = 0..k-2))/(1-x);
%F A126684 Generating function. Define V(k) := [4^(k-1) + Sum ( j=0 to k-2, 4^j 
               * x^(2^j)/(1+x^(2^j)) )] * (1-x^(2^(k-1)))/(1-x) and T(k) := (x^(2^k-1) 
               * V(k), U(k) := x^(3*2^(k-1)-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.
%e A126684 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
%Y A126684 Cf. A000695.
%Y A126684 Sequence in context: A039895 A105425 A032937 this_sequence A089653 A114652 
               A067943
%Y A126684 Adjacent sequences: A126681 A126682 A126683 this_sequence A126685 A126686 
               A126687
%K A126684 easy,nonn
%O A126684 1,3
%A A126684 Jonathan H. B. Deane (J.Deane(AT)surrey.ac.uk), Feb 15 2007, May 04 2007