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 A054490 Expansion of (1+5*x)/(1-6*x+x^2). 17
 1, 11, 65, 379, 2209, 12875, 75041, 437371, 2549185, 14857739, 86597249, 504725755, 2941757281, 17145817931, 99933150305, 582453083899, 3394785353089, 19786259034635, 115322768854721, 672150354093691, 3917579355707425, 22833325780150859 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A Pellian-related second-order recursive sequence. Third binomial transform of 1,8,8,64,64,512. - Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009 Binomial transform of A164607. - R. J. Mathar, Oct 26 2011 Pisano period lengths: 1, 1, 4, 2, 6, 4, 3, 2, 12, 6, 12, 4, 14, 3, 12, 2, 8, 12, 20, 6, ... - R. J. Mathar, Aug 10 2012 From Wolfdieter Lang, Feb 26 2015: (Start) This sequence gives all positive solutions x = x1 = a(n) of the first class of the (generalized) Pell equation x^2 - 2*y^2 = -7. For the corresponding y1 terms see 2*A038723(n). All positive solutions of the second class are given by (x2(n), y2(n)) = (A255236(n), A038725(n+1)), n >= 0. See (A254938(1), 2*A255232(1)) for the fundamental solution (1, 2) of the first class. See the Nagell reference, Theorem 111, p. 210, Theorem 110, p. 208, Theorem 108a, pp. 206-207. This sequence gives also all positive solutions y = y1 of the first class of the Pell equation x^2 - 2*y^2 = 14. The corresponding solutions x1 are given in 4*A038723. This follows from the preceding comment. (End) From Wolfdieter Lang, Mar 19 2015: (Start) a(0) = -(2*A038761(0) - A038762(0)), a(n) = 2*A253811((n-1) + A101386(n-1), for n >= 1. This follows from the general trivial fact that if X^2 - D*Y^2 = N (X, Y positive integers, D > 1, not a square, and N a non-vanishing integer) then x:= D*Y +/- X and y:= Y +/- X (correlated signs) satisfy x^2 - D*y^2 = -(D-1)*N. with integers x and y. Here D = 2 and N = 7. (End) REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196. T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193. Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See y(n) in Lemma 7.9 p. 21. E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (6,-1). FORMULA a(n) = 6*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=11. a(n) = sqrt(8*A038723(n)^2 - 7). a(n) = (11*((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) - ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)). a(n) = 11*S(n, 6) + 5*S(n-1, 6), n >= 0, with Chebyshev's polynomials S(n, x) (A049310) evaluated at x=6: S(n, 6) = A001109(n-1). See the g.f. and the Pell equation comments above. - Wolfdieter Lang, Feb 26 2015 a(n) = 2*A253811(n-1) + A101386(n-1), for n >= 1. See the Mar 19 2015 comment above. - Wolfdieter Lang, Mar 19 2015 From G. C. Greubel, Jan 20 2020: (Start) a(n) = Pell(2*n+1) + 3*Pell(2*n). a(n) = ChebyshevU(n,3) + 5*ChebyshevU(n-1,3). E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x) ). (End) EXAMPLE n = 2: sqrt(8*23^2-7) = 65. 2*19 + 27  = 65. - Wolfdieter Lang, Mar 19 2015 MAPLE a[0]:=1: a[1]:=11: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..30); # Zerinvary Lajos, Jul 26 2006 MATHEMATICA CoefficientList[Series[(1+5x)/(1-6x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *) LinearRecurrence[{6, -1}, {1, 11}, 30] (* G. C. Greubel, Jul 26 2018 *) PROG (MAGMA) I:=[1, 11]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015 (PARI) my(x='x+O('x^30)); Vec((1+5*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jul 26 2018 (Sage) [lucas_number1(2*n+1, 2, -1) + 3*lucas_number1(2*n, 2, -1) for n in (0..30)] # G. C. Greubel, Jan 20 2020 (GAP) a:=[1, 11];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 20 2020 CROSSREFS Cf. A000129, A001109, A038723, A038725, A054488, A054489, A101386, A253811, A255236. Sequence in context: A266765 A036601 A125321 * A126479 A260151 A139611 Adjacent sequences:  A054487 A054488 A054489 * A054491 A054492 A054493 KEYWORD nonn,easy AUTHOR Barry E. Williams, May 04 2000 EXTENSIONS More terms from James A. Sellers, May 05 2000 More terms from Vincenzo Librandi, Mar 20 2015 STATUS approved

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Last modified September 21 13:00 EDT 2020. Contains 337272 sequences. (Running on oeis4.)