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%I #21 Nov 03 2017 08:27:00
%S 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,
%T 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,
%U 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
%N Count of indices x in [0,n] that satisfy the equation A000120(x) + A000120(n-x) = A000120(n) + 2.
%C For every solution x, binomial(n,x) is 4 times an odd integer.
%H Alois P. Heinz, <a href="/A163577/b163577.txt">Table of n, a(n) for n = 0..10000</a>
%H V. Shevelev, <a href="http://arXiv.org/abs/0907.3302">Binomial predictors</a>, arXiv:0907.3302 [math.NT], 2009.
%H L. Spiegelhofer, M. Wallner, <a href="https://arxiv.org/abs/1710.10884">Divisibility of binomial coefficients by powers of two</a>, arXiv:1710.10884
%e For n=8, there are a(8)=2 solutions, namely x=2 and x=6.
%e For n=9, there are a(9)=4 solutions, namely x=2, 3, 6 and 7.
%p read("transforms") ; A000120 := proc(n) wt(n) ; end:
%p 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:
%p seq(A163577(n),n=0..130) ; # _R. J. Mathar_, Jul 08 2009
%t 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 *)
%Y Cf. A000120, A007814.
%Y 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
%K nonn,look
%O 0,5
%A _Vladimir Shevelev_, Jul 31 2009
%E Extended beyond a(22), examples added by _R. J. Mathar_, Jul 08 2009
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