A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n >= 0.
1, 677, 459005, 311204713, 210996336409, 143055204880589, 96991217912702933, 65759902689607707985, 44585117032336113310897, 30228643588021195217080181, 20494975767561338021067051821, 13895563341762999157088244054457
Offset: 0
Examples
(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.
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).
Programs
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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
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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
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Mathematica
LinearRecurrence[{678, -1},{1, 677},11] (* Ray Chandler, Aug 12 2015 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-678*x+x^2)) \\ 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
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
Comments