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A098252
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Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property.
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3
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1, 364, 132131, 47963189, 17410505476, 6319965524599, 2294130074923961, 832762897231873244, 302290637565095063611, 109730668673232276217549, 39831930437745751171906676, 14458881018233034443125905839
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OFFSET
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0,2
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COMMENTS
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(19*a(n))^2 - 365*b(n)^2 = -4 with b(n)=A098253(n) give all positive solutions of this Pell equation.
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LINKS
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FORMULA
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a(n) = S(n, 363) + S(n-1, 363) = S(2*n, sqrt(365)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 363)=A098251(n).
a(n) = (-2/19)*i*((-1)^n)*T(2*n+1, 19*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-363*x+x^2).
a(n) = 363*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=364. - Philippe Deléham, Nov 18 2008
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EXAMPLE
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All positive solutions of Pell equation x^2 - 365*y^2 = -4 are (19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),(911300591=19*47963189,47699653), ...
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MATHEMATICA
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LinearRecurrence[{363, -1}, {1, 364}, 20] (* Harvey P. Dale, Feb 03 2015 *)
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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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