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A218085 Let S_5(x) denote the difference in counts of multiples of 5 in the interval [0,x), those with even digit sums in base 4 in one set, those with odd digit sums in base 4 in the other. Then a(n) = (-1)^s_4(n) *(S_5(n) -10*S_5(floor(n/16)) +5*S_5(floor(n/256))), where s_4(n) = A053737(n). 2
0, -1, 1, -1, -1, 1, -2, 2, 2, -2, 2, -3, -3, 3, -3, 3, 6, -6, 6, -6, -6, 5, -5, 5, 5, -5, 4, -4, -4, 4, -4, 3, -3, 3, -3, 3, 4, -4, 4, -4, -4, 3, -3, 3, 3, -3, 2, -2, 2, -2, 2, -3, -3, 3, -3, 3, 4, -4, 4, -4, -4, 3, -3, 3, 3, -3, 2, -2, -2, 2, -2, 1, 1, -1, 1, -1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The sequence S_5(n) starts 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ... for n >= 0. Apart from the initial 0, these are blocks of 5 repetitions of 1, 2, 3, 4, 5, 6, 7, 6, 7, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...

Theorem. The sequence is periodic with period 2560.

The theorem allows us to write a recursion for S_5(n), considering n modulo 2560: S_5(n) = 10*S_5(floor(n/16)) - 5*S_5(floor(n/256)) + (-1)^s_4(n)*a(n).

LINKS

Peter J. C. Moses, Table of n, a(n) for n = 0..2559

Vladimir Shevelev and Peter J. C. Moses, A family of digit functions with large periods, arXiv:1209.5705 [math.NT], 2012.

FORMULA

-9 <= a(n) <= 9, all 19 values are actually achieved.

EXAMPLE

a(n)=-9 for n=2411, 2412, 2414, 2491, 2492, 2494 (mod 2560);

a(n)=9 for n=2413, 2415, 2493, 2495 (mod 2560).

MAPLE

S := proc(n, j, x)

    a := 0 ;

    for r from j to x-1 by n do

        add(d, d=convert(r, base, n-1)) ;

        a := a+(-1)^% ;

    end do:

    a ;

end proc:

A218085 := proc(n)

    S(5, 0, n)-10*S(5, 0, floor(n/16))+5*S(5, 0, floor(n/256)) ;

    %*(-1)^A053737(n) ;

end proc:

seq(A218085(n), n=0..80) ; # R. J. Mathar, Oct 31 2012

CROSSREFS

Cf. A214458, A217971, A053737.

Sequence in context: A034258 A184349 A290573 * A290726 A090663 A111890

Adjacent sequences:  A218082 A218083 A218084 * A218086 A218087 A218088

KEYWORD

sign,base,easy

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Oct 20 2012

STATUS

approved

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Last modified June 20 09:51 EDT 2021. Contains 345162 sequences. (Running on oeis4.)