A005377 - OEIS

%I M0504 #32 Sep 12 2023 12:02:28

%S 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,

%T 40,42,44,46,49,52,55,58,61,64,67,70,73,76,79,82,85,88,91,94,97,100,

%U 103,106,109,112,115,118,121,124,127,130,133,136,139

%N Number of low discrepancy sequences in base 4.

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

%H 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 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

%F 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.]

%p N := proc(b,n)

%p     option remember;

%p     local d;

%p     add(b^d*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;

%p     %/n ;

%p end proc:

%p M := proc(b,n)

%p     local h;

%p     if n = 0 then

%p         0;

%p     else

%p         add(N(b,h),h=1..n) ;

%p     end if;

%p end proc:

%p nMax := proc(b,s)

%p     local n;

%p     for n from 0 do

%p         if M(b,n) > s then

%p             return n-1 ;

%p         end if;

%p     end do:

%p end proc:

%p A005377 := proc(s)

%p     local n,b;

%p     b := 4 ;

%p     n := nMax(b,s) ;

%p     n*(s-M(b,n))+add( (h-1)*N(b,h),h=1..n) ;

%p end proc:

%p seq(A005377(n),n=1..40) ; # _R. J. Mathar_, Jun 09 2016

%t Np[b_, n_] := Np[b, n] = Sum[b^d*MoebiusMu[n/d], {d, Divisors[n]}]/n;

%t M[b_, n_] := If[n == 0, 0, Sum[Np[b, h], {h, 1, n}]];

%t nMax[b_, s_] := Module[{n}, For[n = 0, True, n++, If[M[b, n] > s, Return[n - 1]]]];

%t 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}]];

%t Table[a[n], {n, 1, 61}] (* _Jean-François Alcover_, Sep 12 2023, after _R. J. Mathar_ *)

%Y Cf. A005356 (base 2), A005357 (base 3), A005358 (base 5), A274039 (Plouffe's g.f.)

%Y 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)).

%K nonn,easy

%O 1,6

%A _N. J. A. Sloane_, _Simon Plouffe_

%E Terms, offset, and formula corrected by _Sean A. Irvine_, Jun 07 2016