A097736 Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n >= 0.
1, 257, 66305, 17106433, 4413393409, 1138638393089, 293764292023553, 75790048703683585, 19553538801258341377, 5044737220675948391681, 1301522649395593426712321, 335787798806842428143387137, 86631950569515950867567169025, 22350707459136308481404186221313
Offset: 0
Examples
(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.
Links
- Indranil Ghosh, Table of n, a(n) for n = 0..413
- Tanya Khovanova, Recursive Sequences
- Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (258,-1).
Programs
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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
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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
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Mathematica
LinearRecurrence[{258, -1},{1, 257},20] (* Ray Chandler, Aug 12 2015 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-258*x+x^2)) \\ G. C. Greubel, Aug 01 2019 -
Sage
((1-x)/(1-258*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
Formula
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.
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.
G.f.: (1-x)/(1-258*x+x^2).
a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=257. - Philippe Deléham, Nov 18 2008
Comments