A214458 - OEIS

%I #19 Feb 20 2021 07:55:13

%S 0,-1,-1,1,1,-1,-1,0,0,0,1,-1,1,-2,-2,2,0,0,0,-1,1,-1,0,0,0,-1,-1,1,1,

%T -1,-1,0,0,0,1,-1,1,-2,-2,2,0,0,0,-1,1,-1,0,0,0,-1,-1,1,1,-1,-1,0,0,0,

%U 1,-1,1,-2,-2,2,0,0,0,-1,1,-1,0,0,0,-1,-1,1,1

%N Let S_3(n) denote difference between multiples of 3 in interval [0,n) with even and odd binary digit sums. Then a(n)=(-1)^A000120(n)*(S_3(n)-3*S_3(floor(n/4))).

%C In 1969, D. J. Newman (see the reference) proved L. Moser's conjecture that difference between numbers of multiples of 3 with even and odd binary digit sums in interval [0,x] is always positive. This fact is known as Moser-Newman phenomenon.

%C Theorem: The sequence is periodic with period of length 24.

%H J. Coquet, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002099551">A summation formula related to the binary digits</a>, Inventiones Mathematicae 73 (1983), pp. 107-115.

%H D. J. Newman, <a href="http://dx.doi.org/10.1090/S0002-9939-1969-0244149-8">On the number of binary digits in a multiple of three</a>, Proc. Amer. Math. Soc. 21 (1969) 719-721.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0709.0885">Two algorithms for evaluation of the Newman digit sum, and a new proof of Coquet's theorem</a>, arXiv:0709.0885 [math.NT], 2007-2012.

%F Recursion for evaluation S_3(n): S_3(n)=3*S_3(floor(n/4))+(-1)^A000120(n)*a(n). As a corollary, we have |S_3(n)-3*S_3(n/4)|<=2.

%Y Cf. A091042, A212500.

%K sign,base

%O 0,14

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jul 18 2012