A005357 Number of low discrepancy sequences in base 3. (Formerly M0637)

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

1,5

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..58.

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

Crossrefs

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

Sequence in context: A336119 A213511 A092919 * A324697 A082664 A342008

Adjacent sequences: A005354 A005355 A005356 * A005358 A005359 A005360

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

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

Status

approved