A042011 Denominators of continued fraction convergents to sqrt(528).
1, 1, 45, 46, 2069, 2115, 95129, 97244, 4373865, 4471109, 201102661, 205573770, 9246348541, 9451922311, 425130930225, 434582852536, 19546776441809, 19981359294345, 898726585392989, 918707944687334, 41321876151635685, 42240584096323019, 1899907576389848521
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Eric Weisstein's World of Mathematics, Lehmer Number
- Index entries for linear recurrences with constant coefficients, signature (0,46,0,-1).
Programs
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Magma
I:=[1,1,45,46]; [n le 4 select I[n] else 46*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 12 2014
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Mathematica
Denominator[Convergents[Sqrt[528], 20]] (* Harvey P. Dale, Nov 14 2011 *) CoefficientList[Series[(1 + x - x^2)/(x^4 - 46 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 12 2014 *)
Formula
G.f.: -(x^2-x-1) / (x^4-46*x^2+1). - Colin Barker, Nov 29 2013
a(n) = 46*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Jan 12 2014
From Peter Bala, May 27 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(11) + sqrt(12) and beta = sqrt(11) - sqrt(12) be the roots of the equation x^2 - sqrt(44)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 44 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 44*a(2*n) + a(2*n - 1). (End)
a(2*n) = A041241(2*n) = numerator of continued fraction [4,11,4,11,...,4,11] with n pairs of 4,11. - Greg Dresden, Aug 10 2021
Extensions
More terms from Colin Barker, Nov 29 2013
Comments