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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))).
2
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, -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, 1, -1, 1, -2, -2, 2, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1
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.
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
Sequence in context: A237452 A132784 A180834 * A133873 A163326 A028953
KEYWORD
sign,base
AUTHOR
STATUS
approved