%I #47 Jul 06 2023 20:54:12

%S 1,39,1481,56239,2135601,81096599,3079535161,116941239519,

%T 4440687566561,168629186289799,6403468391445801,243163169688650639,

%U 9233796979777278481,350641122061847931639,13315128841370444123801,505624254850015028772799,19200406555459200649242561,729109824852599609642444519,27686972937843325965763649161

%N Pell equation solutions (3*a(n))^2 - 10*b(n)^2 = -1 with b(n) = A097315(n), n >= 0.

%H Indranil Ghosh, <a href="/A097314/b097314.txt">Table of n, a(n) for n = 0..631</a>

%H Christian Aebi and Grant Cairns, <a href="http://math.colgate.edu/~integers/x48/x48.pdf">Lattice equable quadrilaterals III: tangential and extangential cases</a>, Integers (2023) Vol. 23, #A48.

%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 (38,-1).

%F G.f.: (1 + x)/(1 - 38*x + x^2).

%F a(n) = S(n, 38) + S(n-1, 38) = S(2*n, 2*sqrt(10)), with Chebyshev polynomials of the second 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, 3*i)/(3*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

%F a(n) = ((3 + sqrt(10))*(19 + 6*sqrt(10))^n - ((-3 + sqrt(10))*(19 - 6*sqrt(10))^n))/6. - _Gerry Martens_, Jul 09 2015

%F a(n) = (1/3)*sinh((2*n + 1)*arcsinh(3)). - _Bruno Berselli_, Apr 03 2018

%e (x,y) = (3,1), (117,37), (4443,1405), ... give the positive integer solutions to x^2 - 10*y^2 = -1.

%t LinearRecurrence[{38, -1}, {1, 39}, 20] (* _Ray Chandler_, Aug 11 2015 *)

%o (PARI) Vec((1+x)/(1-38*x+x^2) + O(x^20)) \\ _Michel Marcus_, Jul 10 2015

%Y Cf. A078987 for S(n, 38).

%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

%E More terms from _Indranil Ghosh_, Feb 04 2017