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 1-2 of 2 results.

A347051 a(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).

Original entry on oeis.org

1, 2, 13, 158, 3173, 95348, 4007789, 224531532, 16170278093, 1455549559902, 160126621867313, 21138169636045218, 3297714589844921321, 600205193521411725640, 126046388354086307305721, 30251733410174235165098680, 8228597533955746051214146681, 2517981097123868465906693983066
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2021

Keywords

Comments

a(n) is the denominator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].

Examples

			a(1) =    2 because 1/(1*2)                               = 1/2.
a(2) =   13 because 1/(1*2 + 1/(2*3))                     = 6/13.
a(3) =  158 because 1/(1*2 + 1/(2*3 + 1/(3*4)))           = 73/158.
a(4) = 3173 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.

Crossrefs

Cf. A001040, A001046, A002378, A036246, A071896, A102038, A346960, A347052.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[n_] := a[n] = n (n + 1) a[n - 1] + a[n - 2]; Table[a[n], {n, 0, 17}]
Table[Denominator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]

Formula

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9478401587876967481571909904361736371398357108358019737901443045685048723... - Vaclav Kotesovec, Aug 14 2021

A347052 Decimal expansion of the continued fraction 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5 + 1/(5*6 + ...))))).

Original entry on oeis.org

4, 6, 2, 0, 2, 3, 3, 2, 5, 0, 8, 0, 2, 3, 8, 6, 1, 8, 5, 0, 3, 5, 5, 9, 1, 4, 9, 4, 1, 7, 5, 7, 1, 9, 1, 5, 9, 7, 7, 0, 3, 0, 2, 3, 9, 4, 2, 0, 4, 4, 7, 4, 7, 3, 8, 5, 0, 3, 9, 3, 2, 6, 6, 0, 3, 5, 6, 0, 7, 7, 8, 9, 8, 1, 5, 2, 4, 1, 3, 0, 8, 3, 7, 2, 9, 8, 4, 1, 1, 0, 8, 2, 1, 2, 5, 5, 5, 4, 3, 3, 5, 6, 3, 7, 8, 9
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2021

Keywords

Examples

			0.46202332508023861850355914941757191597703...

Crossrefs

Cf. A002378, A052119, A073823, A073824, A346960, A347051.

Programs

  • Mathematica
    terms=106; RealDigits[ContinuedFractionK[k(k+1),{k,terms}],10,terms][[1]] (* Stefano Spezia, Aug 23 2025 *)

Formula

Equals lim_{n -> oo} A346960(n)/A347051(n).
Showing 1-2 of 2 results.

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