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.

A269353 Expansion of 1/(1-2x/(1-3x/(1-5x/(1-7x/(1-11x/...))))) where 2,3,5,7,11,... are the prime numbers.

Original entry on oeis.org

1, 2, 10, 80, 910, 14180, 294820, 7898900, 262166950, 10401023300, 480352145980, 25403319156500, 1524514142067820, 103259722304891000, 7855753426977689320, 667222680003329539400, 62774912079259300426030, 6482982674137778455277720, 728040412975153836997647580
Offset: 0

Views

Author

Benedict W. J. Irwin, Feb 25 2016

Keywords

Links

Crossrefs

Cf. A000040, A285407.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,
     `if`(x>0, b(x-1, y)*ithprime(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 12 2023
  • Mathematica
    CoefficientList[Series[1/(1+ContinuedFractionK[-Prime[k]*x,1,{k,1,50}]),{x,0,50}],x]

Formula

G.f.: 1/(1-2x/(1-3x/(1-5x/(1-7x/(1-11x/(1-13x/(...))))))), by definition.

Extensions

Offset corrected by Vaclav Kotesovec, Aug 26 2017

A292801 Expansion of 1/(1 + x^2 + x^3/(1 + x^5 + x^7/(1 + x^11 + x^13/(1 + ... + x^prime(2*k)/(1 + x^prime(2*k+1) + ...))))), a continued fraction.

Original entry on oeis.org

1, 0, -1, -1, 1, 2, 0, -3, -1, 3, 4, -3, -7, -1, 11, 6, -10, -17, 8, 26, 8, -40, -28, 33, 71, -19, -99, -49, 141, 125, -99, -285, 30, 371, 253, -492, -541, 263, 1122, 57, -1352, -1197, 1672, 2260, -548, -4345, -871, 4804, 5387, -5475, -9182, 294, 16526, 5725, -16587, -23366
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

Crossrefs

Cf. A000040, A092869, A111374, A285407, A292802, A292803.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 + x^2 + ContinuedFractionK[x^Prime[2 k], 1 + x^Prime[2 k + 1], {k, 1, nmax}]), {x, 0, nmax}], x]

A292802 Expansion of 1/(1 - x^2 - x^3/(1 - x^5 - x^7/(1 - x^11 - x^13/(1 - ... - x^prime(2*k)/(1 - x^prime(2*k+1) - ...))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 10, 11, 17, 25, 31, 50, 64, 93, 134, 178, 266, 360, 512, 731, 1001, 1447, 2003, 2829, 4011, 5575, 7939, 11097, 15634, 22085, 30909, 43724, 61369, 86389, 121810, 171007, 241216, 339125, 477292, 672364, 945252, 1331677, 1873473, 2636704, 3712653
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

Crossrefs

Cf. A000040, A088352, A285407, A292800, A292801.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[1/(1 - x^2 + ContinuedFractionK[-x^Prime[2 k], 1 - x^Prime[2 k + 1], {k, 1, nmax}]), {x, 0, nmax}], x]

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 1-4 of 4 results.

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