login
a(n) = 6*a(n-1) - a(n-2), with a(1) = 13, a(2) = 73.
4

%I #67 Dec 23 2023 08:32:29

%S 13,73,425,2477,14437,84145,490433,2858453,16660285,97103257,

%T 565959257,3298652285,19225954453,112057074433,653116492145,

%U 3806641878437,22186734778477,129313766792425,753695865976073,4392861429064013,25603472708408005,149227974821384017

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

%C a(n) is the area of the 4-generalized Fibonacci snowflake.

%C a(n) is the area of the 5-generalized Fibonacci snowflake, for n >= 2.

%C From _Wolfdieter Lang_, Feb 07 2015: (Start)

%C 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.

%C The other part of the proper solutions are given in (A254758(n), A254759(n)) for n >= 0.

%C 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)

%C The terms of this sequence are hypotenuses of Pythagorean triangles whose difference between legs is equal to 7. - _César Aguilera_, Sep 29 2023

%H Vincenzo Librandi, <a href="/A220414/b220414.txt">Table of n, a(n) for n = 1..1000</a>

%H César Aguilera, <a href="https://doi.org/10.31219/osf.io/7etk8">Notes on Perfect Numbers</a>, OSF Preprints, 2023, p. 19.

%H R. De Castro, J. Ramírez, and G. Rubiano, <a href="http://arxiv.org/abs/1212.1368"> Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv:1212.1368 [cs.DM], 2012-2014.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F a(n) = A078343(n)^2 + A078343(n+1)^2 = A060569(2*n-1).

%F G.f.: (13-5*x)/(x^2-6*x+1). - _Harvey P. Dale_, Jan 26 2013

%F From _Wolfdieter Lang_, Feb 07 2015: (Start)

%F 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).

%F a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = 5 and a(1) = 13.

%F a(n) = irrational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1. (End)

%e From _Wolfdieter Lang_, Feb 07 2015: (Start)

%e Pell equation x^2 - 2*y^2 = -7^2 instance:

%e A254757(3)^2 - 2*a(3)^2 = 601^2 - 2*425^2 = -49. (End)

%p with(orthopoly): a := n -> `if`(n=1,13,13*U(n-1,3)-5*U(n-2,3)):

%p seq(a(n),n=1..22); # (after _Wolfdieter Lang_) _Peter Luschny_, Feb 07 2015

%t t = {13, 73}; Do[AppendTo[t, 6*t[[-1]] - t[[-2]]], {30}]; t (* _T. D. Noe_, Dec 20 2012 *)

%t LinearRecurrence[{6,-1},{13,73},40] (* _Harvey P. Dale_, Jan 26 2013 *)

%o (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

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

%K nonn,easy

%O 1,1

%A _José Luis Ramírez Ramírez_, Dec 13 2012