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.

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 9, 11, 15, 23, 28, 43, 57, 78, 113, 149, 214, 293, 403, 569, 774, 1086, 1502, 2072, 2896, 3986, 5548, 7691, 10636, 14797, 20459, 28400, 39386, 54542, 75724, 104886, 145468, 201733, 279545, 387786, 537472, 745233, 1033383, 1432415, 1986394
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2017

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 5*x^10 + ...

Links

Crossrefs

Cf. A000040, A206739, A206741, A206742, A206743, A227310, A269353.

Programs

  • Maple
    R:= 1:
for i from numtheory:-pi(50) to 1 by -1 do
  R:= series(1-x^ithprime(i)/R, x, 51);
od:
R:= series(1/R, x, 51):
seq(coeff(R,x,j),j=0..50); # Robert Israel, Apr 20 2017
  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * d^n, where d = 1.3864622092472465020397266918102624708859968795203700659786636158522760956... and c = 0.15945087310540003725148530084775272562567007586487061850065597143186... - Vaclav Kotesovec, Aug 25 2017

A285380 G.f.: 1/(1 - 1!*x/(1 - 2!*x/(1 - 3!*x/(1 - 4!*x/(1 - 5!*x/(1 - 6!*x/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 21, 459, 48069, 31721355, 151932395493, 5929991210130219, 2103657835595933507013, 7506346835525189003011779147, 295743497615320848280307669164734117, 140189609286888251994538844205855399795958635
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2017

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 459*x^4 + 48069*x^5 + 31721355*x^6 + 151932395493*x^7 + ...

Links

Crossrefs

Cf. A000142, A001147, A268646, A269353.

Programs

  • Mathematica
    nmax = 12; CoefficientList[Series[1/(1 + ContinuedFractionK[-k! x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
  • PARI
    a(n) = my(A=1+O(x)); for(i=1, n, A=1-(n-i+1)!*x/A); polcoef(1/A, n); \\ Seiichi Manyama, Apr 15 2021

Formula

a(n) ~ A000178(n) ~ BarnesG(n+2) ~ exp(1/12 - n - 3*n^2/4) * n^(5/12 + n + n^2/2) * (2*Pi)^((n+1)/2) / A, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Aug 26 2017

A292803 Expansion of 1/(1 + x^2/(1 + x^3/(1 + x^5/(1 + x^7/(1 + x^11/(1 + ... + x^prime(k)/(1 + ... ))))))), a continued fraction.

Original entry on oeis.org

1, 0, -1, 0, 1, 1, -1, -2, 0, 3, 1, -3, -3, 3, 5, -1, -8, -1, 9, 6, -11, -11, 8, 21, -5, -27, -8, 38, 20, -36, -50, 38, 72, -9, -118, -15, 131, 100, -170, -166, 118, 330, -94, -411, -129, 618, 294, -567, -817, 663, 1124, -139, -1963, -162, 2087, 1691, -2902, -2605, 1851, 5562, -1758
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

Crossrefs

Cf. A000040, A269353, A285407, A292801.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 + ContinuedFractionK[x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Showing 1-3 of 3 results.

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