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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A054765 a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600
Offset: 0

Views

Author

N. J. A. Sloane, May 26 2000

Keywords

Comments

The denominators of the convergents of [1/3, 4/5, 9/7, 16/9, ...]. To produce Pi the above continued fraction is used. It is formed by n^2/(2*n+1) which starts at n=1. Most numerators of continued fractions are 1 & thus are left out of the brackets. In the case of Pi they vary. Therefore here both numerators & denominators are given. The first 4 convergents are 1/3,5/19,44/160,476/1744. The value of this continued fraction is .273239... . 4*INV(1+.273239...) is Pi. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008
Starting with offset 1 = row sums of triangle A155729. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]

Links

Crossrefs

Cf. A155729, A012244, A054766.

Programs

  • Maple
    A054765 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        (2*n-1)*procname(n-1)+(n-1)^2*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Jul 13 2013
  • Mathematica
    RecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],
a[0] == 0, a[1] == 1}, a, {n,0,50}] (* G. C. Greubel, Feb 18 2017 *)

Formula

a(n) ~ Pi * (1+sqrt(2))^(n + 1/2) * n^n / (2^(9/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

Extensions

More terms from James Sellers, May 27 2000

A155728 INVERTi transform of A054765: (1, 3, 19, 160, 1744, ...).

Original entry on oeis.org

1, 2, 14, 121, 1383, 19108, 309708, 5751027, 120357325, 2803145494, 71926499002, 2016492639229, 61338391284387, 2012321446421976, 70833707268623448, 2663117961930477847, 106515148705020928105, 4516063573152118802282, 202328834841437929100838
Offset: 1

Views

Author

Gary W. Adamson and Alexander R. Povolotsky, Jan 25 2009

Keywords

Comments

This sequence convolved with A054765 prefaced with a 1: (1, 1, 3, 19, 160, ...) = (1, 3, 19, 160, ...), equivalent to row sums of triangle A155729 = A054765.

Examples

			We write (1, 3, 19, 160, ...) in reverse: (..., 19, 3, 1), top row.
Bottom row = (1, 2, ...), so that the format for a(3) = 14 becomes: ...3, 1 = A054765: (1, 3, 19, 160, ...). ..., 1, 2 for current format, take dot product = (3*1 + 1*2) = 5, then subtract from next term in A054765, getting (19 - 5) = 14. So a(3) = 14.
Continuing with analogous operations, we get (1, 2, 14, 121, 1383, ...).

Crossrefs

Cf. A054765, A155729.

Formula

INVERTi transform of A054765 starting with offset 1: (1, 3, 19, 160, 1774, 23184, ...).

Extensions

Corrected by R. J. Mathar, Apr 04 2012
More terms from Alois P. Heinz, Mar 31 2016
Showing 1-2 of 2 results.

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

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