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A098249
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Chebyshev polynomials S(n,291) + S(n-1,291) with Diophantine property.
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4
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1, 292, 84971, 24726269, 7195259308, 2093795732359, 609287362857161, 177300528795701492, 51593844592186277011, 15013631475797410908709, 4368915165612454388157308, 1271339299561748429542867919
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OFFSET
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0,2
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COMMENTS
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(17*a(n))^2 - 293*b(n)^2 = -4 with b(n)=A098250(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) = (-2/17)*i*((-1)^n)*T(2*n+1, 17*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangleA053120.
G.f.: (1+x)/(1-291*x+x^2).
a(n) = S(n, 291) + S(n-1, 291) = S(2*n, sqrt(293)), 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, 227)=A098245(n).
a(n) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=292. [Philippe Deléham,Nov 18 2008]
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EXAMPLE
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All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909),...
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MATHEMATICA
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LinearRecurrence[{291, -1}, {1, 292}, 20] (*Harvey P. Dale,Jan 01 2020 *)
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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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