|
%I #38 Jan 22 2020 02:28:38
%S 1,403,162005,65125607,26180332009,10524428342011,4230794013156413,
%T 1700768668860536015,683704774087922321617,274847618414675912754019,
%U 110488058897925629004794021,44415924829347688184014442423,17855091293338872724344801060025
%N Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n) = A097742(n), n >= 0.
%H Indranil Ghosh, <a href="/A097741/b097741.txt">Table of n, a(n) for n = 0..383</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (402,-1).
%F G.f.: (1 + x)/(1 - 2*201*x + x^2).
%F a(n) = S(n, 2*201) + S(n-1, 2*201) = S(2*n, 2*sqrt(101)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
%F a(n) = ((-1)^n)*T(2*n+1, 10*i)/(10*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F a(n) = 402*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=403. - _Philippe Deléham_, Nov 18 2008
%F a(n) = (1/10)*sinh((2*n + 1)*arcsinh(10)). - _Bruno Berselli_, Apr 03 2018
%F Let h = (10 + sqrt(101))^(2*n+1) then a(n) = (h-1/h)/20 and a(n) = floor(h/20). - _Peter Luschny_, Apr 05 2018
%e (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.
%t LinearRecurrence[{402, -1}, {1, 403}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 402 x + x^2), {x, 0, 20}], x] (* _Harvey P. Dale_, Apr 20 2011 *)
%t a[n_] := Floor[(10 + Sqrt[101])^(2 n + 1)]/20;
%t Table[a[n], {n, 0, 11}] (* _Peter Luschny_, Apr 05 2018 *)
%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*201*x+x^2)) \\ _Altug Alkan_, Apr 05 2018
%Y Cf. A097740 for S(n, 2*201).
%Y Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
| 