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A078989
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Chebyshev sequence with Diophantine property.
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4
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1, 67, 4421, 291719, 19249033, 1270144459, 83810285261, 5530208682767, 364909962777361, 24078527334623059, 1588817894122344533, 104837902484740116119, 6917712746098725319321, 456464203340031130959067
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| One fourth of bisection (even part) of A041024.
(4*a(n))^2 - 17*A078988(n)^2= -1 (Pell -1 equation, see A077232-3).
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)= 66*a(n-1) - a(n-2), n>=1, a(-1)=-1, a(0)=1.
G.f.: (1+x)/(1-66*x+x^2).
a(n)= S(2*n, 2*sqrt(17)) = -i*((-1)^n)*T(2*n+1, 4*i)/4 = S(n, 66) + S(n-1, 66) with i^2=-1 and S(n, x), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n)=A041024(2*n)/4.
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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
| Cf. A097316 for S(n, 66).
Sequence in context: A069397 A103727 A120663 * A156121 A144940 A191941
Adjacent sequences: A078986 A078987 A078988 * A078990 A078991 A078992
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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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