A097733 Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n)=A097732(n), n >= 0. Note that D=50=2*5^2 is not squarefree.
1, 197, 39005, 7722793, 1529074009, 302748930989, 59942759261813, 11868363584907985, 2349876047052519217, 465263588952813896981, 92119840736610099083021, 18239263202259846804541177
Offset: 0
Examples
(x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.
Links
- Indranil Ghosh, Table of n, a(n) for n = 0..434
- 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 sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (198,-1).
Programs
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GAP
a:=[1,197];; for n in [3..20] do a[n]:=198*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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Magma
I:=[1,197]; [n le 2 select I[n] else 198*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
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Mathematica
LinearRecurrence[{198, -1},{1, 197},20] (* Ray Chandler, Aug 11 2015 *) -
PARI
my(x='x+O('x^20)); Vec((1-x)/(1-198*x+x^2)) \\ G. C. Greubel, Aug 01 2019 -
Sage
((1-x)/(1-198*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
Formula
a(n) = S(n, 2*99) - S(n-1, 2*99) = T(2*n+1, 5*sqrt(2))/(5*sqrt(2)), 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, 14*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-198*x+x^2).
a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=197. - Philippe Deléham, Nov 18 2008
a(n) = k^n + k^(-n) - a(n-1) = A003499(3n) - a(n-1), where k = (sqrt(2)+1)^6 = 99 + 70*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011
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.
The aerated sequence (b(n))n>=1 = [1, 0, 197, 0, 39005, 0, 7722793, 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 = -200, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)
Sum_{n >= 1} 1/( a(n) - 1/a(n) ) = 1/196. - Peter Bala, Mar 26 2015
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