A097767 Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n >= 0.
1, 485, 235709, 114554089, 55673051545, 27056988496781, 13149640736384021, 6390698340894137425, 3105866244033814404529, 1509444603902092906463669, 733586971630173118726938605, 356521758767660233608385698361
Offset: 0
Examples
(x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..372
- 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 (486,-1).
Programs
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GAP
a:=[1,485];; for n in [3..20] do a[n]:=486*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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Magma
I:=[1,485]; [n le 2 select I[n] else 486*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
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Mathematica
LinearRecurrence[{486, -1},{1, 485},20] (* Ray Chandler, Aug 12 2015 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-486*x+x^2)) \\ G. C. Greubel, Aug 01 2019 -
Sage
((1-x)/(1-486*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
Formula
a(n) = S(n, 2*243) - S(n-1, 2*243) = T(2*n+1, sqrt(122))/sqrt(122), 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) = ((-1)^n)*S(2*n, 22*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-486*x+x^2).
a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=485. - Philippe Deléham, Nov 18 2008