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A078988
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Chebyshev sequence with Diophantine property.
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8
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1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Bisection (even part) of A041025.
(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - Clark Kimberling, Sep 22 2011
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: (1-x)/(1-66*x+x^2).
a(n)=T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
a(n)= A041025(2*n).
a(n)=66*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=65. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| (x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
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CROSSREFS
| Row 66 of array A094954.
Cf. A097316 for S(n, 66).
Row 4 of array A188647.
Sequence in context: A206888 A188772 A189062 * A027535 A110900 A084272
Adjacent sequences: A078985 A078986 A078987 * A078989 A078990 A078991
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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