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.

A157760 a(n) = 2809*n^2 - 1000*n + 89.

Original entry on oeis.org

1898, 9325, 22370, 41033, 65314, 95213, 130730, 171865, 218618, 270989, 328978, 392585, 461810, 536653, 617114, 703193, 794890, 892205, 995138, 1103689, 1217858, 1337645, 1463050, 1594073, 1730714, 1872973, 2020850, 2174345
Offset: 1

Views

Author

Vincenzo Librandi, Mar 06 2009

Keywords

Comments

The identity (15780962*n^2-5618000*n+500001)^2-(2809*n^2-1000*n+89)*( 297754*n-53000)^2=1 can be written as A157762(n)^2-a(n)*A157761(n)^2=1.

Links

Crossrefs

Subsequence of A031396, see A157757.
Cf. A157761, A157762.

Programs

  • Magma
    I:=[1898, 9325, 22370]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];
  • Mathematica
    LinearRecurrence[{3,-3,1},{1898,9325,22370},50]
  • PARI
    a(n) = 2809*n^2 - 1000*n + 89;

Formula

G.f.: x*(1898 + 3631*x + 89*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A157758 a(n) = 297754*n - 244754.

Original entry on oeis.org

53000, 350754, 648508, 946262, 1244016, 1541770, 1839524, 2137278, 2435032, 2732786, 3030540, 3328294, 3626048, 3923802, 4221556, 4519310, 4817064, 5114818, 5412572, 5710326, 6008080, 6305834, 6603588, 6901342, 7199096
Offset: 1

Views

Author

Vincenzo Librandi, Mar 06 2009

Keywords

Comments

The identity (15780962*n^2-25943924*n+10662963)^2-(2809*n^2-4618*n+1898)*(297754*n-244754)^2=1 can be written as A157759(n)^2-A157757(n)*a(n)^2=1.

Links

Crossrefs

Cf. A157757, A157759.

Programs

  • Magma
    I:=[53000, 350754]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..40]];
  • Mathematica
    LinearRecurrence[{2, -1}, {53000, 350754}, 30]
  • PARI
    a(n) = 297754*n - 244754;

Formula

a(n) = 2*a(n-1) - a(n-2).
G.f.: x*(53000 + 244754*x)/(1-x)^2.

A157759 a(n) = 15780962*n^2 - 25943924*n + 10662963.

Original entry on oeis.org

500001, 21898963, 74859849, 159382659, 275467393, 423114051, 602322633, 813093139, 1055425569, 1329319923, 1634776201, 1971794403, 2340374529, 2740516579, 3172220553, 3635486451, 4130314273, 4656704019, 5214655689, 5804169283
Offset: 1

Views

Author

Vincenzo Librandi, Mar 06 2009

Keywords

Comments

The identity (15780962*n^2 - 25943924*n + 10662963)^2 - (2809*n^2 - 4618*n+1898)*(297754*n - 244754)^2 = 1 can be written as a(n)^2 - A157757(n)*A157758(n)^2 = 1.
This is the case s=53 and r=2309 of the identity (2*(s^2*n-r)^2+1)^2 - (((s^2*n-r)^2+1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2+1)/s^2 is an integer if r^2 == -1 (mod s^2). Therefore, for s=53, nonnegative r values are: 500, 2309, 3309, 5118, 6118, 7927, 8927, 10736, 11736, ... - Bruno Berselli, Apr 24 2018

Links

Crossrefs

Cf. A157757, A157758.

Programs

  • Magma
    I:=[500001, 21898963, 74859849]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..30]];
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {500001, 21898963, 74859849}, 30]
  • PARI
    a(n) = 15780962*n^2 - 25943924*n + 10662963;

Formula

G.f.: x*(500001 - 20398960*x - 10662963*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Showing 1-3 of 3 results.

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

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