A163577 Count of indices x in [0,n] that satisfy the equation A000120(x) + A000120(n-x) = A000120(n) + 2.

0, 0, 0, 0, 2, 0, 1, 0, 2, 4, 1, 0, 5, 2, 2, 0, 2, 4, 5, 8, 5, 2, 4, 0, 5, 10, 4, 4, 10, 4, 4, 0, 2, 4, 5, 8, 9, 10, 12, 16, 5, 10, 6, 4, 12, 8, 8, 0, 5, 10, 12, 20, 12, 8, 12, 8, 10, 20, 8, 8, 20, 8, 8, 0, 2, 4, 5, 8, 9, 10, 12, 16, 9, 18, 14, 20, 20, 24, 24, 32, 5, 10, 14, 20, 14, 12, 16, 8, 12, 24

Offset: 0

Author

Vladimir Shevelev, Jul 31 2009

Keywords

• nonn
• look

Comments

For every solution x, binomial(n,x) is 4 times an odd integer.

Examples

			For n=8, there are a(8)=2 solutions, namely x=2 and x=6.
For n=9, there are a(9)=4 solutions, namely x=2, 3, 6 and 7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• V. Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
• L. Spiegelhofer, M. Wallner, Divisibility of binomial coefficients by powers of two, arXiv:1710.10884

Crossrefs

Cf. A000120, A007814.

A001316 and A163000 count binomial coefficients with 2-adic valuation 0 and 1. A275012 gives a measure of complexity of these sequences. - Eric Rowland, Mar 15 2017

Programs

Maple

read("transforms") ; A000120 := proc(n) wt(n) ; end:
A163577 := proc(n) local a,x ; a := 0 ; for x from 0 to n do if A000120(x)+A000120(n-x) = A000120(n)+2 then a := a+1; fi; od: a; end:
seq(A163577(n),n=0..130) ; # R. J. Mathar, Jul 08 2009

Mathematica

a120[n_] := DigitCount[n, 2, 1]; a[n_] := Count[Range[0, n], x_ /; a120[x] + a120[n-x] == a120[n]+2]; Array[a, 90, 0] (* Jean-François Alcover, Jul 10 2017 *)

Extensions

Extended beyond a(22), examples added by R. J. Mathar, Jul 08 2009