login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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
Sequence in context: A034258 A184349 A290573 * A290726 A090663 A111890
KEYWORD
sign,base,easy
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)