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.

A254757 Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (-1, 5).

Original entry on oeis.org

17, 103, 601, 3503, 20417, 118999, 693577, 4042463, 23561201, 137324743, 800387257, 4664998799, 27189605537, 158472634423, 923646201001, 5383404571583, 31376781228497, 182877282799399, 1065886915567897, 6212444210607983
Offset: 1

Views

Author

Wolfdieter Lang, Feb 07 2015

Keywords

Comments

The corresponding y solutions are given in A220414.
The other part of the proper (sometimes called primitive) solutions are given in (A254758(n), A254759(n)) for n >= 1.
The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 0.

Examples

			The first pairs of positive solutions of this part of the Pell equation  x^2 - 2*y^2 = - 7^2 are: [17, 13], [103, 73], [601, 425], [3503, 2477], [20417, 14437], [118999, 84145], [693577, 490433], [4042463, 2858453], [23561201, 16660285], [137324743, 97103257], ...

References

  • T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Links

Crossrefs

Cf. A049310, A220414, A254758, A254759, A001653, A002315.

Programs

  • Mathematica
    LinearRecurrence[{6,-1},{17,103},20] (* Harvey P. Dale, Sep 01 2017 *)
  • PARI
    Vec((17 + x)/(1 - 6*x + x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015

Formula

a(n) = rational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1.
G.f.: (17 + x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = -1 and a(1) = 17.
a(n) = 17*S(n-1, 6) + S(n-2, 6), n >= 1, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).

A254758 Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).

Original entry on oeis.org

1, 23, 137, 799, 4657, 27143, 158201, 922063, 5374177, 31322999, 182563817, 1064059903, 6201795601, 36146713703, 210678486617, 1227924205999, 7156866749377, 41713276290263, 243122790992201, 1417023469662943
Offset: 0

Views

Author

Wolfdieter Lang, Feb 07 2015

Keywords

Comments

The corresponding y solutions are given in A254759.
The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.
The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 1.

Examples

			The first pairs of positive solutions of this part of the Pell equation  x^2 - 2*y^2 = - 7^2 are: [1, 5], [23, 17], [137, 97], [799, 565], [4657, 3293], [27143, 19193], [158201, 111865], [922063, 651997], [5374177, 3800117], ...

References

  • T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Links

Crossrefs

Cf. A254759, A254757, A220414, A001653, A002315, A049310.

Programs

  • Maple
    with(orthopoly): a := n -> `if`(n=0,1, U(n,3)+17*U(n-1, 3)):
seq(a(n), n=0..19); # Peter Luschny, Feb 07 2015
  • Mathematica
    LinearRecurrence[{6, -1}, {1, 23}, 20] (* Jean-François Alcover, Jun 28 2019 *)
  • PARI
    Vec((1+17*x)/(1-6*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015

Formula

a(n) = rational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.
G.f.: (1+17*x)/(1-6*x+x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = -17 and a(0) = 1.
a(n) = S(n, 6) + 17*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).

A254759 Part of the positive proper solutions y of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).

Original entry on oeis.org

5, 17, 97, 565, 3293, 19193, 111865, 651997, 3800117, 22148705, 129092113, 752403973, 4385331725, 25559586377, 148972186537, 868273532845, 5060669010533, 29495740530353, 171913774171585, 1001986904499157
Offset: 0

Views

Author

Wolfdieter Lang, Feb 07 2015

Keywords

Comments

The corresponding x solutions are given in A254758.
The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.
The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 1.

Examples

			A254758(3)^2 - 2*a(3)^2 = 799^2 - 2*565^2 = -49.
See also A254758 for the first pairs of solutions.

References

  • T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Links

Crossrefs

Cf. A254758, A254757, A220414, A001653, A002315, A049310.

Programs

  • PARI
    Vec((5-13*x)/(1-6*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015

Formula

a(n) = irrational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.
G.f.: (5-13*x)/(1-6*x+x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 13 and a(0) = 5.
a(n) = 5*S(n, 6) - 13*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
Showing 1-3 of 3 results.

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