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

1,1

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

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.

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)

Example

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A066110 A020527 A146618 * A139157 A228027 A159832

Adjacent sequences: A220411 A220412 A220413 * A220415 A220416 A220417

Keyword

nonn,easy

Author

José Luis Ramírez Ramírez, Dec 13 2012

Status

approved