login
A158634
a(n) = 46*n^2 - 1.
2
45, 183, 413, 735, 1149, 1655, 2253, 2943, 3725, 4599, 5565, 6623, 7773, 9015, 10349, 11775, 13293, 14903, 16605, 18399, 20285, 22263, 24333, 26495, 28749, 31095, 33533, 36063, 38685, 41399, 44205, 47103, 50093, 53175, 56349, 59615, 62973, 66423, 69965, 73599
OFFSET
1,1
COMMENTS
The identity (46*n^2 - 1)^2 - (529*n^2 - 23)*(2*n)^2 = 1 can be written as a(n)^2 - A158633(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(-45 - 48*x + x^2)/(x-1)^3.
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(46))*Pi/sqrt(46))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(46))*Pi/sqrt(46) - 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {45, 183, 413}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
PROG
(Magma) I:=[45, 183, 413]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 17 2012
(PARI) for(n=1, 40, print1(46*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 17 2012
CROSSREFS
Sequence in context: A271737 A280887 A158630 * A091197 A184539 A146302
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
EXTENSIONS
Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
STATUS
approved