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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
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1,3
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COMMENTS
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Previous name was: 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".
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. Adams-Watters, Jul 27 2015
Or, base 2 representation Sum{d(i)*2^(m-i): i=0,1,...,m} has even d(i) for all odd i.
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LINKS
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FORMULA
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G.f.: sum(i>=1, T(i, x) + U(i, x) ), where
T := (k,x) -> x^(2^k-1)*V(k,x);
U := (k,x) -> 2*x^(3*2^(k-1)-1)*V(k,x); and
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);
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.
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EXAMPLE
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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
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MATHEMATICA
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nmax = 100;
b[n_] := FromDigits[IntegerDigits[n, 2], 4];
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PROG
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(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 !! (n-1)
a126684_list = tail $ m a000695_list $ map (* 2) a000695_list where
m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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