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.

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A001052 a(n) = n*(n-1)*a(n-1)/2 + a(n-2), a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 3, 11, 69, 701, 10584, 222965, 6253604, 225352709, 10147125509, 558317255704, 36859086001973, 2875567025409598, 261713458398275391, 27482788698844325653, 3298196357319717353751, 448582187384180404435789, 68636372866136921596029468
Offset: 0

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Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001046.

Programs

  • GAP
    a:=[1,2];; for n in [3..20] do a[n]:=Binomial(n-1,2)*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Sep 20 2019
  • Magma
    I:=[1,2]; [n le 2 select I[n] else Binomial(n-1,2)*Self(n-1) + Self(n-2): n in [1..20]]; // G. C. Greubel, Sep 20 2019
  • Maple
    a := proc (n) option remember;
if n < 2 then n+1
else binomial(n,2)*a(n-1)+a(n-2) fi;
end proc;
seq(a(n), n = 0..20); # G. C. Greubel, Sep 20 2019
  • Mathematica
    t = {1, 2}; Do[AppendTo[t, n*(n-1)*t[[-1]]/2 + t[[-2]]], {n, 2, 20}] (* T. D. Noe, Jun 25 2012 *)
  • PARI
    a(n)=if(n<2,max(0,n+1),n*(n-1)*a(n-1)/2+a(n-2))
  • Sage
    def a(n):
    if (n<2): return n+1
    else: return binomial(n,2)*a(n-1)+a(n-2)
[a(n) for n in (0..20)] # G. C. Greubel, Sep 20 2019

Extensions

More terms from James Sellers, Sep 19 2000

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
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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