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 A220414 a(n) = 6*a(n-1) - a(n-2), with a(1) = 13, a(2) = 73. 4
 13, 73, 425, 2477, 14437, 84145, 490433, 2858453, 16660285, 97103257, 565959257, 3298652285, 19225954453, 112057074433, 653116492145, 3806641878437, 22186734778477, 129313766792425, 753695865976073, 4392861429064013, 25603472708408005, 149227974821384017 (list; graph; refs; listen; history; text; internal format)
 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) LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 R. De Castro, J. Ramírez, G. Rubiano, Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv:1212.1368, (2012) Index entries for linear recurrences with constant coefficients, signature (6, -1). 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 a(n) = 1/4*(sqrt(2)*((3-2*sqrt(2))^n - (3+2*sqrt(2))^n) + 10*((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)). - Paolo P. Lava, Feb 01 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

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Last modified February 17 12:10 EST 2019. Contains 320219 sequences. (Running on oeis4.)