A097742 Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n)=A097741(n), n >= 0.
1, 401, 161201, 64802401, 26050404001, 10472197606001, 4209797387208401, 1692328077460171201, 680311677341601614401, 273483601963246388818001, 109939727677547706703222001, 44195497042772214848306426401, 17766479871466752821312480191201, 7142080712832591861952768730436401
Offset: 0
Examples
(x,y) = (10*1=10;1), (4030=10*403;401), (1620050=10*162005;161201), ... give the positive integer solutions to x^2 - 101*y^2 =-1.
Links
- Indranil Ghosh, Table of n, a(n) for n = 0..383
- 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 (402,-1).
Programs
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GAP
a:=[1,401];; for n in [3..20] do a[n]:=402*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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Magma
I:=[1,401]; [n le 2 select I[n] else 402*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
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Mathematica
LinearRecurrence[{402, -1},{1, 401},12] (* Ray Chandler, Aug 12 2015 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-402*x+x^2)) \\ G. C. Greubel, Aug 01 2019 -
Sage
((1-x)/(1-402*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
Formula
a(n) = S(n, 2*201) - S(n-1, 2*201) = T(2*n+1, sqrt(101))/sqrt(101), 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, 20*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-402*x+x^2).
a(n) = 402*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=401. - Philippe Deléham, Nov 18 2008
Comments