A126684 Union of A000695 and 2*A000695.

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

Offset: 1

Author

Jonathan Deane, Feb 15 2007, May 04 2007

Keywords

• easy
• nonn
• base

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 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.

Union of A000695 and 2*A000695. - Ralf Stephan, May 05 2004

Union of A000695 and A062880. - Franklin T. Adams-Watters, Aug 30 2014

Integers, the binary representation of which contains no pair of 1 bits whose difference in bit index is odd. - Frederik P.J. Vandecasteele, May 29 2025

Examples

			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
		

Links

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

Crossrefs

Cf. A000695, A062880, A033053, A032937.

Programs

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'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Dec 03 2011

Mathematica

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

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,",")))

Python

from gmpy2 import digits
def A126684(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x):
        s = digits(x,4)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return int(s,2)
        return int(s[:i]+'1'*(l-i),2)
    def f(x): return n-1+x-g(x)-g(x>>1)
    return bisection(f,n-1,n-1) # Chai Wah Wu, Oct 29 2024

Formula

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.

Extensions

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