A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0.
1, 101, 10301, 1050601, 107151001, 10928351501, 1114584702101, 113676711262801, 11593909964103601, 1182465139627304501, 120599850332020955501, 12300002268726510156601, 1254479631559772015017801, 127944622416828019021659101, 13049097006884898168194210501
Offset: 0
Examples
(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- 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 (102,-1).
Programs
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GAP
a:=[1,101];; for n in [3..20] do a[n]:=102*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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Magma
I:=[1,101]; [n le 2 select I[n] else 102*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
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Mathematica
LinearRecurrence[{102,-1},{1,101},20] (* Harvey P. Dale, Apr 12 2014 *) CoefficientList[Series[(1-x)/(1-102x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Apr 13 2014 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-102*x+x^2)) \\ G. C. Greubel, Aug 01 2019 -
Sage
((1-x)/(1-102*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
Formula
a(n) = S(n, 2*51) - S(n-1, 2*51) = T(2*n+1, sqrt(26))/sqrt(26), 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, 10*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-102*x+x^2).
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=101. - Philippe Deléham, Nov 18 2008
Extensions
More terms from Harvey P. Dale, Apr 12 2014
Comments