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A097736
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Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n >= 0.
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5
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1, 257, 66305, 17106433, 4413393409, 1138638393089, 293764292023553, 75790048703683585, 19553538801258341377, 5044737220675948391681, 1301522649395593426712321, 335787798806842428143387137, 86631950569515950867567169025, 22350707459136308481404186221313
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.
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MATHEMATICA
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LinearRecurrence[{258, -1}, {1, 257}, 20] (* Ray Chandler, Aug 12 2015 *)
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PROG
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(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-258*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(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
(Sage) ((1-x)/(1-258*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(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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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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