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Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n >= 0.
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%I #35 Sep 08 2022 08:45:14

%S 1,577,333505,192765313,111418017409,64399421297089,37222754091700033,

%T 21514687465581321985,12435452132351912407297,

%U 7187669817811939790095681,4154460719243168846762896321,2401271108052733781489163977857

%N Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n >= 0.

%H Seiichi Manyama, <a href="/A097770/b097770.txt">Table of n, a(n) for n = 0..362</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (578,-1).

%F a(n) = S(n, 2*289) - S(n-1, 2*289) = T(2*n+1, sqrt(145))/sqrt(145), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

%F a(n) = ((-1)^n)*S(2*n, 24*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

%F G.f.: (1-x)/(1-578*x+x^2).

%F a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=577. - _Philippe Deléham_, Nov 18 2008

%e (x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 =-1.

%t LinearRecurrence[{578, -1},{1, 577},12] (* _Ray Chandler_, Aug 12 2015 *)

%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-578*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019

%o (Magma) I:=[1,577]; [n le 2 select I[n] else 578*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019

%o (Sage) ((1-x)/(1-578*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019

%o (GAP) a:=[1,577];; for n in [3..20] do a[n]:=578*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019

%Y Cf. A097768 for S(n, 486).

%Y Row 12 of array A188647.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 31 2004