|
%I #33 Sep 08 2022 08:45:14
%S 1,257,66305,17106433,4413393409,1138638393089,293764292023553,
%T 75790048703683585,19553538801258341377,5044737220675948391681,
%U 1301522649395593426712321,335787798806842428143387137,86631950569515950867567169025,22350707459136308481404186221313
%N Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n >= 0.
%H Indranil Ghosh, <a href="/A097736/b097736.txt">Table of n, a(n) for n = 0..413</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 (258,-1).
%F a(n) = S(n, 2*129) - S(n-1, 2*129) = T(2*n+1, sqrt(65))/sqrt(65), 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, 16*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F G.f.: (1-x)/(1-258*x+x^2).
%F a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=257. - _Philippe Deléham_, Nov 18 2008
%e (x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.
%t LinearRecurrence[{258, -1},{1, 257},20] (* _Ray Chandler_, Aug 12 2015 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-258*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,257]; [n le 2 select I[n] else 258*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-258*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,257];; for n in [3..20] do a[n]:=258*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Cf. A097734 for S(n, 258).
%Y Row 8 of array A188647.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
| 