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URL: https://oeis.org/A158740

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A158740
a(n) = 72*n^2 + 1.
2
1, 73, 289, 649, 1153, 1801, 2593, 3529, 4609, 5833, 7201, 8713, 10369, 12169, 14113, 16201, 18433, 20809, 23329, 25993, 28801, 31753, 34849, 38089, 41473, 45001, 48673, 52489, 56449, 60553, 64801, 69193, 73729, 78409, 83233, 88201, 93313, 98569, 103969, 109513
OFFSET
0,2
COMMENTS
The identity (72*n^2 + 1)^2 - (1296*n^2 + 36)*(2*n)^2 = 1 can be written as a(n)^2 - A158739(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: -(1 + 70*x + 73*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 22 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)) + 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(1 + 72*x + 72*x^2).
a(n) = A157889(2*n) for n > 0. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 73, 289}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
PROG
(Magma) I:=[1, 73, 289]; [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(72*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012
CROSSREFS
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