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.

A237132 Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.

Original entry on oeis.org

3, 4, 5, 9, 12, 23, 31, 60, 81, 157, 212, 411, 555, 1076, 1453, 2817, 3804, 7375, 9959, 19308, 26073, 50549, 68260, 132339, 178707, 346468, 467861, 907065, 1224876, 2374727, 3206767, 6217116, 8395425, 16276621, 21979508, 42612747, 57543099, 111561620
Offset: 1

Views

Author

Colin Barker, Feb 04 2014

Keywords

Comments

The corresponding values of y are given by a(n+2).
Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 704 = 0.

Examples

			9 is in the sequence because (x, y) = (9, 23) is a solution to x^2 - 3xy + y^2 + 11 = 0.

Links

Crossrefs

Cf. A001519, A005248, A055819, A237133, A218735.

Programs

  • PARI
    Vec(-x*(x-1)*(3*x^2+7*x+3)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

Formula

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(3*x^2+7*x+3) / ((x^2-x-1)*(x^2+x-1)).
a(n) = F(n+2) + (-1)^n*F(n-3), n>1, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014
Let h(n) = hypergeom([(1 - n)/2, n mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 3. - Peter Luschny, Sep 03 2019

A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

64, 256, 320, 576, 704, 1024, 1216, 1280, 1600, 1856, 1984, 2304, 2624, 2816, 2880, 3136, 3520, 3776, 3904, 4096, 4544, 4864, 5056, 5120, 5184, 5696, 6080, 6336, 6400, 6464, 6976, 7424, 7744, 7936, 8000, 8384, 8896, 9216, 9280, 9536, 9664, 9920, 10496, 10816
Offset: 1

Views

Author

Colin Barker, Feb 16 2014

Keywords

Examples

			64 is in the sequence because x^2 - 18xy + y^2 + 64 = 0 has integer solutions, for example (x, y) = (1, 13).

Crossrefs

Cf. A001519 (n = 64), A052995 (n = 256), A055819 (n = 256), A005248 (n = 320), A237132 (n = 704), A237133 (n = 1216).
Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330.

A218735 Values of x in the solutions to x^2 - 3xy + y^2 + 29 = 0, where 0 < x < y.

Original entry on oeis.org

5, 6, 9, 13, 22, 33, 57, 86, 149, 225, 390, 589, 1021, 1542, 2673, 4037, 6998, 10569, 18321, 27670, 47965, 72441, 125574, 189653, 328757, 496518, 860697, 1299901, 2253334, 3403185, 5899305, 8909654, 15444581, 23325777, 40434438, 61067677, 105858733
Offset: 1

Views

Author

Colin Barker, Feb 05 2014

Keywords

Comments

The corresponding values of y are given by a(n+2).
Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1856 = 0.

Examples

			13 is in the sequence because (x, y) = (13, 33) is a solution to x^2 - 3xy + y^2 + 29 = 0.

Links

Crossrefs

Cf. A001519, A005248, A055819, A237132, A237133.

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-1},{5,6,9,13},40] (* Harvey P. Dale, Nov 30 2024 *)
  • PARI
    Vec(-x*(x-1)*(5*x^2+11*x+5)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

Formula

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(5*x^2+11*x+5) / ((x^2-x-1)*(x^2+x-1)).
Showing 1-3 of 3 results.

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

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