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A005357 Number of low discrepancy sequences in base 3.

%I A005357 M0637 #28 Sep 13 2023 09:43:30
%S A005357 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,
%T A005357 58,61,64,67,70,73,77,81,85,89,93,97,101,105,109,113,117,121,125,129,
%U A005357 133,137,141,145,149,153,157,161,165,169,173,177
%N A005357 Number of low discrepancy sequences in base 3.
%D A005357 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005357 Harald Niederreiter, <a href="http://dx.doi.org/10.1016/0022-314X(88)90025-X">Low-discrepancy and low-dispersion sequences</a>, J. Number Theory 30 (1988), no. 1, 51-70.
%F A005357 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
%t A005357 Np[b_, n_] := Np[b, n] = Sum[b^d*MoebiusMu[n/d], {d, Divisors[n]}]/n;
%t A005357 M[b_, n_] := If[n == 0, 0, Sum[Np[b, h], {h, 1, n}]];
%t A005357 nMax[b_, s_] := Module[{n}, For[n = 0, True, n++, If[M[b, n] > s, Return[n - 1]]]];
%t A005357 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}]];
%t A005357 Table[a[n], {n, 1, 58}] (* _Jean-François Alcover_, Sep 12 2023, after _R. J. Mathar_ in A005356 *)
%Y A005357 Cf. A005356 (base 2), A005377 (base 4), A005358 (base 5).
%K A005357 nonn
%O A005357 1,5
%A A005357 _N. J. A. Sloane_, _Simon Plouffe_
%E A005357 a(33) onwards corrected and incorrect g.f. removed by _Sean A. Irvine_, May 27 2016