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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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4
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1, 257, 66305, 17106433, 4413393409, 1138638393089, 293764292023553, 75790048703683585, 19553538801258341377, 5044737220675948391681, 1301522649395593426712321, 335787798806842428143387137
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| (x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.
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CROSSREFS
| Cf. A097734 for S(n, 258).
Row 8 of array A188647.
Sequence in context: A168116 A086022 A125649 * A103349 A194155 A121237
Adjacent sequences: A097733 A097734 A097735 * A097737 A097738 A097739
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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