This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214458 #19 Feb 20 2021 07:55:13 %S A214458 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 A214458 -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 A214458 1,-1,1,-2,-2,2,0,0,0,-1,1,-1,0,0,0,-1,-1,1,1 %N A214458 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 A214458 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 A214458 Theorem: The sequence is periodic with period of length 24. %H A214458 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 A214458 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 A214458 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 A214458 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 A214458 Cf. A091042, A212500. %K A214458 sign,base %O A214458 0,14 %A A214458 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jul 18 2012