A097732 - OEIS

%I #47 Jul 06 2023 20:54:21

%S 1,199,39401,7801199,1544598001,305822602999,60551330795801,

%T 11988857674965599,2373733268312392801,469987198268178808999,

%U 93055091523831091789001,18424438134520287995413199,3647945695543493192000024401,722274823279477131728009418199

%N Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=2*5^2 is not squarefree.

%C Also numbers k such that (7*k+1)^2 + (7*k-1)^2 is a square. - _Bruno Berselli_, Oct 11 2019

%H Indranil Ghosh, <a href="/A097732/b097732.txt">Table of n, a(n) for n = 0..434</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 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html ">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

%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 (198,-1).

%F G.f.: (1 + x)/(1 - 2*99*x + x^2).

%F a(n) = S(n, 2*99) + S(n-1, 2*99) = S(2*n, 10*sqrt(2)), 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, 7*i)/(7*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

%F a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=199. - _Philippe Deléham_, Nov 18 2008

%F From _Peter Bala_, Mar 23 2015: (Start)

%F a(n) = ( Pell(6*n + 6 - 2*k) + Pell(6*n + 2*k) )/( Pell(6 - 2*k) + Pell(2*k) ), for k an arbitrary integer.

%F a(n) = ( Pell(6*n + 6 - 2*k - 1) - Pell(6*n + 2*k + 1) )/( Pell(6 - 2*k - 1) - Pell(2*k + 1) ), for k an arbitrary integer, k != 1.

%F The aerated sequence (b(n))n>=1 = [1, 0, 199, 0, 39401, 0, 7801199, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -196, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

%F a(n) = (1/7)*sinh((2*n + 1)*arcsinh(7)). - _Bruno Berselli_, Apr 03 2018

%e (x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.

%t LinearRecurrence[{198, -1}, {1, 199}, 12] (* _Ray Chandler_, Aug 11 2015 *)

%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*99*x+x^2)) \\ _Altug Alkan_, Apr 05 2018

%Y Cf. A097731 for S(n, 2*99), A100047.

%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