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.

A157757 a(n) = 2809*n^2 - 4618*n + 1898.

Original entry on oeis.org

89, 3898, 13325, 28370, 49033, 75314, 107213, 144730, 187865, 236618, 290989, 350978, 416585, 487810, 564653, 647114, 735193, 828890, 928205, 1033138, 1143689, 1259858, 1381645, 1509050, 1642073, 1780714, 1924973, 2074850
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-a(n)*A157758(n)^2=1.
From Klaus Purath, Mar 31 2025: (Start)
Numbers k such that k*53^2-1 is a square, and k is the sum of two squares (see FORMULA).
All a(n) = D satisfy the Pell equation (k*x)^2 - D*(53*y)^2 = -1 for any integer n where a(1-n) = A157760(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*53^2 - 1), x(0) = 1, x(1) = 4*D*53^2 - 1, y(0) = 1, y(1) = 4*D*53^2 - 3. The two recurrences are of the form (4*D*53^2 - 2, -1).
It follows from the above that this sequence and A157760 are subsequences of A031396. (End)

Links

Crossrefs

Subsequence of A031396.
Cf. A157758, A157759.

Programs

  • Magma
    I:=[89, 3898, 13325]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
  • Mathematica
    LinearRecurrence[{3,-3,1},{89,3898,13325},40]
Table[2809n^2-4618n+1898,{n,40}] (* Harvey P. Dale, Aug 02 2024 *)
  • PARI
    a(n) = 2809*n^2 - 4618*n + 1898;

Formula

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f.: x*(-89-3631*x-1898*x^2)/(x-1)^3.
a(n) = (28*n - 23)^2 + (45*n - 37)^2. - Klaus Purath, Mar 31 2025
53^2*a(n) - 1 = (2809*n-2309)^2. - Klaus Purath, Mar 31 2025

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.
Showing 1-2 of 2 results.

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