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

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

Offset: 1

Author

José Luis Ramírez Ramírez, Dec 13 2012

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• César Aguilera, Notes on Perfect Numbers, OSF Preprints, 2023, p. 19.
• R. De Castro, J. Ramírez, and G. Rubiano, Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv:1212.1368 [cs.DM], 2012-2014.
• Index entries for linear recurrences with constant coefficients, signature (6,-1).
• Index entries for sequences related to Chebyshev polynomials.

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)