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A097315
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Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n):=A097314(n), n>=0.
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9
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1, 37, 1405, 53353, 2026009, 76934989, 2921503573, 110940200785, 4212806126257, 159975692596981, 6074863512559021, 230684837784645817, 8759948972303982025, 332647376109766671133, 12631840343198829521029
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OFFSET
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0,2
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COMMENTS
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Hypotenuses of primitive Pythagorean triples in A195616 and A195617. - Clark Kimberling, Sep 22 2011
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LINKS
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Table of n, a(n) for n=0..14.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)= S(n, 38) - S(n-1, 38) = T(2*n+1, sqrt(10))/sqrt(10), with Chebyshev polynomials of the second 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, 6*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-38*x+x^2).
a(n)=38*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=37. [From Philippe DELEHAM, Nov 18 2008]
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EXAMPLE
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(x,y) = (3,1), (117,37), (4443,1405),... give the positive integer solutions to x^2 - 10*y^2 =-1.
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CROSSREFS
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Row 3 of array A188647.
Cf. A221874.
Sequence in context: A207185 A189061 A009981 * A158741 A094490 A009695
Adjacent sequences: A097312 A097313 A097314 * A097316 A097317 A097318
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 31 2004
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EXTENSIONS
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Typo in recurrence formula corrected Laurent Bonaventure (bonave(AT)free.fr), Oct 03 2010
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STATUS
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approved
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