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 1-3 of 3 results.

A270124 Ratios of successive terms of A270121.

Original entry on oeis.org

2, 16, 3600, 4354563600, 30582275103386435842563600, 8210597206238423089030527211173884373017314096293353169731842563600
Offset: 0

Views

Author

Andrew Hone, Mar 11 2016

Keywords

Comments

The ratios A270121(n+1)/A270121(n) generate the sequence for n>=1.

Links

Programs

  • Mathematica
    (* b = A270121 *)
b[1] = 7; b[2] = 112;
b[n_] := b[n] = (b[n - 1]^2 (1 + (n - 1) b[n - 1]))/b[n - 2];
a[0] = 2; a[n_] := b[n + 1]/b[n];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Dec 16 2018 *)

Formula

A270121 is generated by a recurrence of second order: if A270121(n)=x(n) then x(n+1)*x(n-1)=x(n)^2*(1+n*x(n)) for n>=1 with x(1)=7,x(2)=112.

A270137 Decimal expansion of the constant 6/A270121(1) + Sum_{n>=2} 1/A270121(n).

Original entry on oeis.org

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

Views

Author

Andrew Hone, Mar 11 2016

Keywords

Comments

A270121 is defined by the following recurrence: if A270121(n)=x(n) then x(n+1)*x(n-1)=x(n)^2*(1+n*x(n)) for n>=1, with x(1)=7, x(2)=112; and for A270124, if A270124(n)=y(n) then y(0)=2 and y(n)=x(n+1)/x(n) for n>=1. Both of these sequences appear in the continued fraction expansion of this number, which is transcendental.

Examples

			0.86607390873015929971... = 6/A270121(1) + Sum_{n>=2} 1/A270121(n) = 6/7 + 1/112 + 1/403200 + 1/1755760043520000 + ... = [0; 1, 6, 2, 7, 32, 112, 10800, 403200, 17418254400, ...] = [0; 1, 6, A270124(0), A270121(1), 2*A270124(1), A270121(2), 3*A270124(2), A270121(3), 4*A270124(3), ...] (continued fraction).

Links

Crossrefs

Cf. A112373, A114550, A114551, A114552.

Formula

The continued fraction expansion takes the form
[0; 1, 6, A270124(0), A270121(1), ..., n*A270124(n-1), A270121(n), (n+1)*A270124(n), A270121(n+1), ...].

Extensions

More terms from Jon E. Schoenfield, Nov 12 2016

A270138 Continued fraction expansion of the constant 6/A270121(1)+Sum_{n>=2}1/A270121(n).

Original entry on oeis.org

0, 1, 6, 2, 7, 32, 112, 10800, 403200, 17418254400, 1755760043520000
Offset: 0

Views

Author

Andrew Hone, Mar 11 2016

Keywords

Comments

A270121 is defined by the following recurrence: if A270121(n)=x(n) then x(n+1)*x(n-1)=x(n)^2*(1+n*x(n)) for n>=1, with x(1)=7, x(2)=112; and for A270124, if A270124(n)=y(n) then y(0)=2 and y(n)=x(n+1)/x(n) for n>=1. Both of these sequences appear in this continued fraction expansion, which defines a transcendental number.

Examples

			6/A270121(1)+Sum_{n>=2}1/A270121(n)=6/7+1/112+1/403200+1/1755760043520000+...
=[0;1,6,2,7,32,112,10800,403200,17418254400,...]
=[0;1,6,A270124(0),A270121(1),2*A270124(1),A270121(2),3*A270124(2),A270121(3),4*A270124(3),...] (continued fraction).

Links

Crossrefs

Cf. A112373, A114550, A114551, A114552.

Formula

a(2*n+1) = n*A270124(n-1), a(2*n+2) = A270121(n) for n>=1.
Showing 1-3 of 3 results.

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