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%I #53 Sep 08 2022 08:45:01
%S 1,11,65,379,2209,12875,75041,437371,2549185,14857739,86597249,
%T 504725755,2941757281,17145817931,99933150305,582453083899,
%U 3394785353089,19786259034635,115322768854721,672150354093691,3917579355707425,22833325780150859
%N Expansion of (1+5*x)/(1-6*x+x^2).
%C A Pellian-related second-order recursive sequence.
%C Third binomial transform of 1,8,8,64,64,512. - Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%C Binomial transform of A164607. - _R. J. Mathar_, Oct 26 2011
%C 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
%C From _Wolfdieter Lang_, Feb 26 2015: (Start)
%C 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.
%C This sequence also gives 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)
%C From _Wolfdieter Lang_, Mar 19 2015: (Start)
%C a(0) = -(2*A038761(0) - A038762(0)), a(n) = 2*A253811((n-1) + A101386(n-1), for n >= 1.
%C 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)
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
%H G. C. Greubel, <a href="/A054490/b054490.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, <a href="https://arxiv.org/abs/2004.04535">Symmetries of biplanes</a>, arXiv:2004.04535 [math.GR], 2020. See y(n) in Lemma 7.9 p. 21.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%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) = 6*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=11.
%F a(n) = sqrt(8*A038723(n)^2 - 7).
%F 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)).
%F 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
%F a(n) = 2*A253811(n-1) + A101386(n-1), for n >= 1. See the Mar 19 2015 comment above. - _Wolfdieter Lang_, Mar 19 2015
%F From _G. C. Greubel_, Jan 20 2020: (Start)
%F a(n) = Pell(2*n+1) + 3*Pell(2*n).
%F a(n) = ChebyshevU(n,3) + 5*ChebyshevU(n-1,3).
%F E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x) ). (End)
%e n = 2: sqrt(8*23^2-7) = 65.
%e 2*19 + 27 = 65. - _Wolfdieter Lang_, Mar 19 2015
%p 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
%t CoefficientList[Series[(1+5x)/(1-6x+x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 20 2015 *)
%t LinearRecurrence[{6, -1}, {1, 11}, 30] (* _G. C. Greubel_, Jul 26 2018 *)
%o (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
%o (PARI) my(x='x+O('x^30)); Vec((1+5*x)/(1-6*x+x^2)) \\ _G. C. Greubel_, Jul 26 2018
%o (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
%o (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
%Y Cf. A000129, A001109, A038723, A038725, A054488, A054489, A101386, A253811, A255236.
%K nonn,easy
%O 0,2
%A _Barry E. Williams_, May 04 2000
%E More terms from _James A. Sellers_, May 05 2000
%E More terms from _Vincenzo Librandi_, Mar 20 2015
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