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URL: https://oeis.org/A158742

⇱ A158742 - OEIS

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A158742
a(n) = 74*n^2 + 1.
2
1, 75, 297, 667, 1185, 1851, 2665, 3627, 4737, 5995, 7401, 8955, 10657, 12507, 14505, 16651, 18945, 21387, 23977, 26715, 29601, 32635, 35817, 39147, 42625, 46251, 50025, 53947, 58017, 62235, 66601, 71115, 75777, 80587, 85545, 90651, 95905, 101307, 106857, 112555
OFFSET
0,2
COMMENTS
The identity (74*n^2 + 1)^2 - (1369*n^2 + 37)*(2*n)^2 = 1 can be written as a(n)^2 - A158741(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: -(1 + 72*x + 75*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 23 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(74))*Pi/sqrt(74) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(74))*Pi/sqrt(74) + 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 75, 297}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
PROG
(Magma) I:=[1, 75, 297]; [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 21 2012
(PARI) for(n=0, 40, print1(74*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012
CROSSREFS
Sequence in context: A348489 A258056 A174685 * A292313 A158765 A226741
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 25 2009
EXTENSIONS
Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009
STATUS
approved