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A078989 Chebyshev sequence with Diophantine property. 5
1, 67, 4421, 291719, 19249033, 1270144459, 83810285261, 5530208682767, 364909962777361, 24078527334623059, 1588817894122344533, 104837902484740116119, 6917712746098725319321, 456464203340031130959067 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

One fourth of bisection (even part) of A041024.

(4*a(n))^2 - 17*A078988(n)^2= -1 (Pell -1 equation, see A077232-3).

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..548

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (66, -1).

Index entries for sequences related to Chebyshev polynomials.

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.

EXAMPLE

(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.

CROSSREFS

Cf. A097316 for S(n, 66).

Sequence in context: A103727 A120663 A261974 * A156121 A280004 A211961

Adjacent sequences:  A078986 A078987 A078988 * A078990 A078991 A078992

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified July 20 16:53 EDT 2017. Contains 289628 sequences.