A163000 Count of integers x in [0,n] satisfying A000120(x) + A000120(n-x) = A000120(n) + 1.

0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 4, 4, 2, 4, 4, 0, 1, 2, 4, 4, 4, 8, 8, 8, 2, 4, 8, 8, 4, 8, 8, 0, 1, 2, 4, 4, 4, 8, 8, 8, 4, 8, 12, 16, 8, 16, 16, 16, 2, 4, 8, 8, 8, 16, 16, 16, 4, 8, 16, 16, 8, 16, 16, 0, 1, 2, 4, 4, 4, 8, 8, 8, 4, 8, 12, 16, 8, 16, 16, 16, 4, 8, 12, 16, 12, 24, 24, 32, 8, 16, 24, 32, 16

Offset: 0

Author

Vladimir Shevelev, Jul 20 2009

Keywords

• nonn
• base

Comments

For every solution x, binomial(n,x) is 2 times an odd integer.

A generalization: for every solution 0 <= x <= n of the equation A000120(x) + A000120(n-x) = A000120(n) + r, binomial(n,x) is 2^r times an odd integer.

Apparently this is also the number of 2's in the n-th row of A034931. - R. J. Mathar, Jul 28 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.
• Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
• L. Spiegelhofer and M. Wallner, Divisibility of binomial coefficients by powers of two, arXiv:1710.10884 [math.NT], 2017.

Crossrefs

Cf. A000120, A007814.

A001316 and A163577 count binomial coefficients with 2-adic valuation 0 and 2. A275012 gives a measure of complexity of these sequences. - Eric Rowland, Mar 15 2017

Programs

Maple

A163000 := proc(n) local a,x; a := 0 ; for x from 0 to n do if A000120(x)+A000120(n-x) = A000120(n)+1 then a := a+1; fi; od: a; end:
seq(A163000(n),n=0..100) ; # R. J. Mathar, Jul 21 2009

Mathematica

okQ[x_, n_] := DigitCount[x, 2, 1] + DigitCount[n - x, 2, 1] == DigitCount[n, 2, 1] + 1; a[n_] := Count[Range[0, n], x_ /; okQ[x, n]]; Table[a[n], {n, 0, 92}] (* Jean-François Alcover, Jul 13 2017 *)

PARI

a(n) = my(z=hammingweight(n)+1); sum(x=0, n, hammingweight(x) + hammingweight(n-x) == z); \\ Michel Marcus, Jun 06 2021

Formula

a(n)=0 iff n=2^k-1, k>=0. a(n)=1 iff n=2^k, k>=1.

Conjecture: a(n) = A033264(n)* 2^(A000120(n)-1); from [Davis & Webb]. - R. J. Mathar, Jul 28 2017

Extensions

Extended beyond a(22) by R. J. Mathar, Jul 21 2009