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A078988 Chebyshev sequence with Diophantine property. 9
1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585, 29220420180900779828304449, 1928104896615996323709378049 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Colin Barker, Table of n, a(n) for n = 0..549

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

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. - Philippe Deléham, Nov 18 2008

EXAMPLE

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

PROG

(PARI) Vec((1-x)/(1-66*x+x^2) + O(x^50)) \\ Colin Barker, Jun 15 2015

CROSSREFS

Row 66 of array A094954.

Cf. A097316 for S(n, 66).

Row 4 of array A188647.

Sequence in context: A188772 A207186 A189062 * A027535 A110900 A084272

Adjacent sequences:  A078985 A078986 A078987 * A078989 A078990 A078991

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified August 20 22:35 EDT 2017. Contains 290837 sequences.