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Search: id:A078988
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| A078988 |
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Chebyshev sequence with Diophantine property. |
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+0
6
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| 1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Bisection (even part) of A041025.
(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
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LINKS
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Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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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
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(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
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Row 66 of array A094954.
Cf. A097316 for S(n, 66).
Sequence in context: A069225 A075474 A133853 this_sequence A027535 A110900 A084272
Adjacent sequences: A078985 A078986 A078987 this_sequence A078989 A078990 A078991
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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