A005377 Number of low discrepancy sequences in base 4. (Formerly M0504)

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

Table of n, a(n) for n=1..61.

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 A321152

Adjacent sequences: A005374 A005375 A005376 * A005378 A005379 A005380

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Terms, offset, and formula corrected by Sean A. Irvine, Jun 07 2016

Status

approved