cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005377 Number of low discrepancy sequences in base 4.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

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

Links

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)).

Programs

  • 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 *)

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.]

Extensions

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

A005356 Number of low discrepancy sequences in base 2.

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 11, 14, 18, 22, 26, 30, 34, 38, 43, 48, 53, 58, 63, 68, 73, 78, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 198, 205, 212, 219, 226, 233, 240, 247, 254, 261, 268, 275, 282, 289, 296
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

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

Links

Crossrefs

Cf. A005357 (base 3), A005377 (base 4), A005358 (base 5).

Programs

  • 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:
A005356 := proc(s)
    local n,b;
    b := 2 ;
    n := nMax(b,s) ;
    n*(s-M(b,n))+add( (h-1)*N(b,h),h=1..n) ;
end proc:
seq(A005356(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 = 2; n = nMax[b, s]; n*(s - M[b, n]) + Sum[(h - 1)*Np[b, h], {h, 1, n}]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 09 2023, after R. J. Mathar *)

Extensions

More terms from Sean A. Irvine, May 27 2016

A005357 Number of low discrepancy sequences in base 3.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

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

Links

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
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.