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.

A220414 a(n) = 6*a(n-1) - a(n-2), with a(1) = 13, a(2) = 73.

Original entry on oeis.org

13, 73, 425, 2477, 14437, 84145, 490433, 2858453, 16660285, 97103257, 565959257, 3298652285, 19225954453, 112057074433, 653116492145, 3806641878437, 22186734778477, 129313766792425, 753695865976073, 4392861429064013, 25603472708408005, 149227974821384017
Offset: 1

Views

Author

José Luis Ramírez Ramírez, Dec 13 2012

Keywords

Comments

a(n) is the area of the 4-generalized Fibonacci snowflake.
a(n) is the area of the 5-generalized Fibonacci snowflake, for n >= 2.
From Wolfdieter Lang, Feb 07 2015: (Start)
This sequence gives one part of the positive proper (sometimes called primitive) solutions y of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0) = (-1, 5). The corresponding x solutions are given in A254757.
The other part of the proper solutions are given in (A254758(n), A254759(n)) for n >= 0.
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. (End)
The terms of this sequence are hypotenuses of Pythagorean triangles whose difference between legs is equal to 7. - César Aguilera, Sep 29 2023

Examples

			From _Wolfdieter Lang_, Feb 07 2015: (Start)
Pell equation x^2 - 2*y^2 = -7^2 instance:
A254757(3)^2 - 2*a(3)^2 = 601^2 - 2*425^2 = -49. (End)

Links

Crossrefs

Cf. A001653, A171587, A078343, A254757, A049310, A254758, A254759.

Programs

  • Magma
    I:=[13, 73]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 01 2013
  • Maple
    with(orthopoly): a := n -> `if`(n=1,13,13*U(n-1,3)-5*U(n-2,3)):
seq(a(n),n=1..22); # (after Wolfdieter Lang) Peter Luschny, Feb 07 2015
  • Mathematica
    t = {13, 73}; Do[AppendTo[t, 6*t[[-1]] - t[[-2]]], {30}]; t (* T. D. Noe, Dec 20 2012 *)
LinearRecurrence[{6,-1},{13,73},40] (* Harvey P. Dale, Jan 26 2013 *)

Formula

a(n) = A078343(n)^2 + A078343(n+1)^2 = A060569(2*n-1).
G.f.: (13-5*x)/(x^2-6*x+1). - Harvey P. Dale, Jan 26 2013
From Wolfdieter Lang, Feb 07 2015: (Start)
a(n) = 13*S(n-1, 6) - 5*S(n-2, 6), n >= 1, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = 5 and a(1) = 13.
a(n) = irrational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1. (End)

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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