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%I #36 Sep 08 2022 08:45:14
%S 1,485,235709,114554089,55673051545,27056988496781,13149640736384021,
%T 6390698340894137425,3105866244033814404529,1509444603902092906463669,
%U 733586971630173118726938605,356521758767660233608385698361
%N Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n >= 0.
%H Seiichi Manyama, <a href="/A097767/b097767.txt">Table of n, a(n) for n = 0..372</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 (486,-1).
%F a(n) = S(n, 2*243) - S(n-1, 2*243) = T(2*n+1, sqrt(122))/sqrt(122), 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, 22*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F G.f.: (1-x)/(1-486*x+x^2).
%F a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=485. - _Philippe Deléham_, Nov 18 2008
%e (x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.
%t LinearRecurrence[{486, -1},{1, 485},20] (* _Ray Chandler_, Aug 12 2015 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-486*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,485]; [n le 2 select I[n] else 486*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-486*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,485];; for n in [3..20] do a[n]:=486*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Cf. A097765 for S(n, 486).
%Y Row 11 of array A188647.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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