A005357 Number of low discrepancy sequences in base 3.

0, 0, 0, 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

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.

Crossrefs

Cf. A005356 (base 2), A005377 (base 4), A005358 (base 5).

Programs

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 = 3; n = nMax[b, s]; n*(s - M[b, n]) + Sum[(h - 1)*Np[b, h], {h, 1, n}]];
Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Sep 12 2023, after R. J. Mathar in A005356 *)

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) = Sum_{h=1..r} (h-1) * N(3, h) + r * (n - M(3, r)) [From Niederreiter paper]. - Sean A. Irvine, May 27 2016

Extensions

a(33) onwards corrected and incorrect g.f. removed by Sean A. Irvine, May 27 2016