The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16. 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. MATHEMATICA CoefficientList[Series[(1-x)/(1-66x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Apr 15 2019 *) LinearRecurrence[{66, -1}, {1, 65}, 21] (* G. C. Greubel, Aug 01 2019 *) PROG (PARI) Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015 (MAGMA) I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019 (Sage) ((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019 (GAP) a:=[1, 65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 21 11:17 EDT 2020. Contains 337268 sequences. (Running on oeis4.)