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A038725 a(n) = 6*a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=2.

%I A038725 #67 Jul 10 2025 11:03:54
%S A038725 1,2,11,64,373,2174,12671,73852,430441,2508794,14622323,85225144,
%T A038725 496728541,2895146102,16874148071,98349742324,573224305873,
%U A038725 3340996092914,19472752251611,113495517416752,661500352248901,3855506596076654,22471539224211023,130973728749189484
%N A038725 a(n) = 6*a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=2.
%C A038725 From _Wolfdieter Lang_, Feb 26 2015: (Start)
%C A038725 The sequence {2*a(n+1)}_{n >= 0}, gives all positive solutions y = y2(n) = 2*a(n+1) of the second class of the Pell equation x^2 - 2*y^2  = -7. For the corresponding terms x = x2(n) see A255236(n).
%C A038725 See A255236 for comments on the first class solutions and the relation to the Pell equation x^2 - 2*y^2 = 14. (End)
%D A038725 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H A038725 Stefano Spezia, <a href="/A038725/b038725.txt">Table of n, a(n) for n = 0..1300</a>
%H A038725 I. Adler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H A038725 Seyed Hassan Alavi, Ashraf Daneshkhah, and Cheryl E Praeger, <a href="https://arxiv.org/abs/2004.04535">Symmetries of biplanes</a>, arXiv:2004.04535 [math.GR], 2020. See x'(n) in Lemma 7.9 p. 21.
%H A038725 Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, <a href="http://dx.doi.org/10.13140/RG.2.2.16511.73129">Lune and Lens Sequences</a>, ResearchGate preprint, 2024. See pp. 55, 56.
%H A038725 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://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 A038725 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A038725 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%H A038725 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A038725 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(n) = ((4-sqrt(2))/8)*(3+2*sqrt(2))^(n-1)+((4+sqrt(2))/8)*(3-2*sqrt(2))^(n-1). - _Antonio Alberto Olivares_, Mar 29 2008
%F A038725 From _Michael Somos_, Sep 28 2008: (Start)
%F A038725 Sequence satisfies -7 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 6*u*v.
%F A038725 G.f.: (1 - 4*x) / (1 - 6*x + x^2). a(n) = (7 + a(n-1)^2) / a(n-2). (End)
%F A038725 From _Wolfdieter Lang_, Feb 26 2015: (Start)
%F A038725 a(n) = S(n, 6) - 4*S(n-1, 6), n>=0, with the Chebyshev polynomials S(n, x) (A049310), with S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n-1). See the g.f. and the Pell equation comment above.
%F A038725 a(n) = 6*a(n-1) - a(n-2), n >= 1, a(-1) = 4, a(0) = 1. (See the name.) (End)
%F A038725 From _Wolfdieter Lang_, Mar 19 2015: (Start)
%F A038725 a(n+1) = sqrt((A255236(n)^2 + 7)/2)/2, n >= 0.
%F A038725 a(n+1) = (A038761(n) + A038762(n))/2, n >= 0. See the Mar 19 2015 comment on A054490. (End)
%F A038725 E.g.f.: exp(3*x)*(4*cosh(2*sqrt(2)*x) - sqrt(2)*sinh(2*sqrt(2)*x))/4. - _Stefano Spezia_, May 01 2020
%e A038725 n = 2: a(3) = sqrt((181^2 + 7)/2)/2 = 64.
%e A038725 a(3) = (53 + 75)/2 = 64. - _Wolfdieter Lang_, Mar 19 2015
%p A038725 a[0]:=1: a[1]:=2: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # _Zerinvary Lajos_, Jul 26 2006
%t A038725 Union[Flatten[NestList[{#[[2]],#[[3]],6#[[3]]-#[[2]]}&,{1,2,11},25]]]  (* _Harvey P. Dale_, Mar 04 2011 *)
%t A038725 LinearRecurrence[{6,-1},{1,2},30] (* _Harvey P. Dale_, Jun 12 2017 *)
%o A038725 (PARI) {a(n) = real((3 + 2*quadgen(8))^n * (1 - quadgen(8) / 4))} /* _Michael Somos_, Sep 28 2008 */
%o A038725 (PARI) {a(n) = polchebyshev(n, 1, 3) - polchebyshev(n-1, 2, 3)} /* _Michael Somos_, Sep 28 2008 */
%Y A038725 Cf. A001653 and A001541. Cf. A001109.
%Y A038725 A038723(n) = a(-n).
%K A038725 easy,nonn
%O A038725 0,2
%A A038725 _Barry E. Williams_, May 02 2000