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 1-4 of 4 results.

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

A005358 Number of low discrepancy sequences in base 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162
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), A005377 (base 4).

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 = 5; n = nMax[b, s]; n*(s - M[b, n]) + Sum[(h - 1)*Np[b, h], {h, 1, n}]];
Table[a[n], {n, 1, 79}] (* Jean-François Alcover, Sep 12 2023, after R. J. Mathar in A005356 *)

Extensions

More terms from Sean A. Irvine, May 27 2016

A274039 Expansion of (x^4 + x^10) / (1 - 2*x + x^2).

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, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112
Offset: 0

Views

Author

Sean A. Irvine, Jun 07 2016

Keywords

Comments

This g.f. was incorrectly conjectured by Plouffe in his 1992 disseration to be the g.f. for A005377.

Links

Crossrefs

Cf. A004279, A005377.

Programs

  • Mathematica
    CoefficientList[Series[(x^4+x^10)/(1-2x+x^2),{x,0,120}],x] (* Harvey P. Dale, Sep 10 2018 *)

Formula

G.f.: z^4*(z^2+1)*(z^4-z^2+1)/(z-1)^2. [Simon Plouffe in his 1992 dissertation.]
a(n) = 2*(n-6), n>=9. - R. J. Mathar, Jun 09 2016
a(n) = A004279(n-4) for n >= 4. - Georg Fischer, Oct 30 2018
Showing 1-4 of 4 results.

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