A060569 -id:A060569 - cp's OEIS frontend

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

José Luis Ramírez Ramírez, Dec 13 2012

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

			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)

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.

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

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

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

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 *)

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)

A117474 The values of 'a' in a^2 + b^2 = c^2 where b - a = 7 and gcd(a,b,c)=1.

Original entry on oeis.org

5, 8, 48, 65, 297, 396, 1748, 2325, 10205, 13568, 59496, 79097, 346785, 461028, 2021228, 2687085, 11780597, 15661496, 68662368, 91281905, 400193625, 532029948, 2332499396, 3100897797, 13594802765, 18073356848

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 19 2006

The values of 'c' are in A060569.

			a(5) = 6*48 - 5 + 14 = 297, 297^2 + 304^2 = 425^2, 304 - 297 = 7, and gcd(297, 304, 425) = 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

Cf. A060569.

g:=proc(n) option remember; if n=1 then RETURN(5) fi; if n=2 then RETURN(8) fi; if n=3 then RETURN(48) fi; if n=4 then RETURN(65) fi; 6*g(n-2)-g(n-4)+14; end; # N. J. A. Sloane, Oct 06 2007

LinearRecurrence[{1,6,-6,-1,1},{5,8,48,65,297},40] (* Harvey P. Dale, May 07 2025 *)

a(n) = 6*a(n-2) - a(n-4) + 14; a(1)=5, a(2)=8, a(3)=48, a(4)=65.

G.f.: x*(3*x^4 + x^3 - 10*x^2 - 3*x-5) / ((x-1)*(x^2-2*x-1)*(x^2+2*x-1)). [Colin Barker, Dec 17 2012]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula