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.

A285484 G.f.: 1/(1 + x/(1 + x^3/(1 + x^6/(1 + x^10/(1 + x^15/(1 + ... + x^(k*(k+1)/2)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -6, 9, -13, 18, -26, 38, -54, 77, -111, 160, -229, 328, -472, 679, -974, 1398, -2010, 2888, -4146, 5954, -8555, 12289, -17647, 25346, -36410, 52297, -75109, 107881, -154961, 222574, -319679, 459167, -659528, 947295, -1360612, 1954295, -2807031, 4031809, -5790982
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2017

Keywords

Examples

			G.f.: A(x) = 1 - x + x^2 - x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 9*x^8 - 13*x^9 + ...

Crossrefs

Cf. A000217, A002105, A206740, A285408.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[1/(1 + ContinuedFractionK[x^(k (k + 1)/2), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * c * d^n, where d = 1.43632929358192465555987661527... and c = 0.4856490524128736949896673... - Vaclav Kotesovec, Aug 26 2017

A295073 Expansion of 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^(k*(k+1)*(2*k+1)/6)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 107, 142, 188, 249, 330, 439, 584, 776, 1030, 1366, 1813, 2408, 3199, 4249, 5642, 7490, 9944, 13204, 17534, 23285, 30920, 41056, 54514, 72384, 96116, 127631, 169478, 225042, 298819, 396783, 526869, 699608, 928981, 1233552
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2017

Keywords

Crossrefs

Cf. A000330, A206739, A285408, A295072.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^(k (k + 1) (2 k + 1)/6), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^A000330(k)/(1 - ...))))))), a continued fraction.
a(n) ~ c * d^n, where d = 1.327852426419013789340602526081665378868516025761586390361772232517175463... and c = 0.366619510178622647108505347089605503045273798338613615745637268621... - Vaclav Kotesovec, Sep 18 2021

A291169 Expansion of 1/(1 + x/(1 + x^8/(1 + x^27/(1 + x^64/(1 + x^125/(1 + ... + x^(k^3)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, 0, -1, 2, -3, 4, -5, 6, -7, 7, -6, 4, -1, -3, 8, -14, 21, -28, 34, -38, 39, -36, 28, -14, -7, 35, -69, 107, -147, 184, -213, 228, -222, 188, -120, 14, 133, -318, 533, -764, 990, -1183, 1309, -1330, 1204, -892, 363, 400, -1393, 2584
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2017

Keywords

Examples

			G.f.: 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^10 + 2*x^11 - 3*x^12 + ...

Crossrefs

Cf. A007325, A285408.

A291170 Expansion of 1 + x/(1 + x^4/(1 + x^9/(1 + x^16/(1 + x^25/(1 + ... + x^(k^2)/(1 + ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 1, -2, 0, 0, -1, 3, -1, 0, 1, -4, 3, 0, -1, 4, -6, 1, 1, -4, 10, -4, -1, 4, -13, 10, 0, -4, 15, -20, 4, 5, -16, 32, -14, -5, 16, -44, 34, 1, -17, 54, -65, 13, 19, -60, 105, -48, -18, 64, -149, 113, 4, -69, 189
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2017

Keywords

Crossrefs

Cf. A003823.

Formula

Convolution inverse of A285408.
Showing 1-4 of 4 results.

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