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A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n>=0. 4
1, 677, 459005, 311204713, 210996336409, 143055204880589, 96991217912702933, 65759902689607707985, 44585117032336113310897, 30228643588021195217080181, 20494975767561338021067051821, 13895563341762999157088244054457 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..353

Tanya Khovanova, Recursive Sequences

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (678, -1).

FORMULA

a(n) = ((-1)^n)*S(2*n, 26*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-678*x+x^2).

a(n) = S(n, 2*339) - S(n-1, 2*339) = T(2*n+1, sqrt(170))/sqrt(170), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = 678*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=677. - Philippe Deléham, Nov 18 2008

EXAMPLE

(x,y) = (13*1=13;1), (8827=13*679;677), (5984693=13*460361;459005), ... give the positive integer solutions to x^2 - 170*y^2 =-1.

MATHEMATICA

LinearRecurrence[{678, -1}, {1, 677}, 11] (* Ray Chandler, Aug 12 2015 *)

PROG

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-678*x+x^2)) \\ G. C. Greubel, Aug 01 2019

(MAGMA) I:=[1, 677]; [n le 2 select I[n] else 678*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

(Sage) ((1-x)/(1-678*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

(GAP) a:=[1, 677];; for n in [3..20] do a[n]:=678*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

CROSSREFS

Cf. A097771 for S(n, 678).

Row 13 of array A188647.

Sequence in context: A108824 A205471 A144381 * A248887 A031524 A158385

Adjacent sequences:  A097770 A097771 A097772 * A097774 A097775 A097776

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)