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⇱ A005377 - OEIS

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A005377
Number of low discrepancy sequences in base 4.
5
0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139
OFFSET
1,6
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Harald Niederreiter, Low-discrepancy and low-dispersion sequences, J. Number Theory 30 (1988), no. 1, 51-70.
FORMULA
Let N(b,n) = (1/n) * Sum_{d|n} mobius(n/d) * b^d. Let M(b,n) = Sum_{k=1..n} N(b,k) with M(b,0) = 0. Let r = r(b,n) be the largest value r such that M(b,r) <= n. Then a(n) = r * (n - M(4, r)) + Sum_{h=1..r} (h-1) * N(4, h) [From Niederreiter paper]. - Sean A. Irvine, Jun 07 2016
G.f.: z^4 * (z^2+1) * (z^4-z^2+1) / (z-1)^2; [Conjectured by Simon Plouffe in his 1992 dissertation, but is incorrect.]
MAPLE
N := proc(b, n)
option remember;
local d;
add(b^d*numtheory[mobius](n/d), d=numtheory[divisors](n)) ;
%/n ;
end proc:
M := proc(b, n)
local h;
if n = 0 then
0;
else
add(N(b, h), h=1..n) ;
end if;
end proc:
nMax := proc(b, s)
local n;
for n from 0 do
if M(b, n) > s then
return n-1 ;
end if;
end do:
end proc:
A005377 := proc(s)
local n, b;
b := 4 ;
n := nMax(b, s) ;
n*(s-M(b, n))+add( (h-1)*N(b, h), h=1..n) ;
end proc:
seq(A005377(n), n=1..40) ; # R. J. Mathar, Jun 09 2016
MATHEMATICA
Np[b_, n_] := Np[b, n] = Sum[b^d*MoebiusMu[n/d], {d, Divisors[n]}]/n;
M[b_, n_] := If[n == 0, 0, Sum[Np[b, h], {h, 1, n}]];
nMax[b_, s_] := Module[{n}, For[n = 0, True, n++, If[M[b, n] > s, Return[n - 1]]]];
a[s_] := Module[{n, b}, b = 4; n = nMax[b, s]; n*(s - M[b, n]) + Sum[(h - 1)*Np[b, h], {h, 1, n}]];
Table[a[n], {n, 1, 61}] (* Jean-François Alcover, Sep 12 2023, after R. J. Mathar *)
CROSSREFS
Cf. A005356 (base 2), A005357 (base 3), A005358 (base 5), A274039 (Plouffe's g.f.)
Cf. A001037 (N(2,n)), A027376 (N(3,n)), A027377 (N(4,n)), A062692 (M(2,n)), A114945 (M(3,n)), A114946 (M(4,n)).
Sequence in context: A004272 A004279 A274039 * A120370 A011866 A392071
KEYWORD
nonn,easy
EXTENSIONS
Terms, offset, and formula corrected by Sean A. Irvine, Jun 07 2016
STATUS
approved