OFFSET
0,14
COMMENTS
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.
Theorem: The sequence is periodic with period of length 24.
LINKS
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Vladimir Shevelev, Two algorithms for evaluation of the Newman digit sum, and a new proof of Coquet's theorem, arXiv:0709.0885 [math.NT], 2007-2012.
FORMULA
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.
CROSSREFS
KEYWORD
sign,base
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Jul 18 2012
STATUS
approved