OFFSET
0,2
COMMENTS
Also numbers k such that (7*k+1)^2 + (7*k-1)^2 is a square. - Bruno Berselli, Oct 11 2019
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..434
Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (198,-1).
FORMULA
G.f.: (1 + x)/(1 - 2*99*x + x^2).
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).
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.
a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=199. - Philippe Deléham, Nov 18 2008
From Peter Bala, Mar 23 2015: (Start)
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.
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.
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)
a(n) = (1/7)*sinh((2*n + 1)*arcsinh(7)). - Bruno Berselli, Apr 03 2018
EXAMPLE
(x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.
MATHEMATICA
LinearRecurrence[{198, -1}, {1, 199}, 12] (* Ray Chandler, Aug 11 2015 *)
PROG
(PARI) x='x+O('x^99); Vec((1+x)/(1-2*99*x+x^2)) \\ Altug Alkan, Apr 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
STATUS
approved