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A158547
a(n) = 24*n^2 + 1.
2
1, 25, 97, 217, 385, 601, 865, 1177, 1537, 1945, 2401, 2905, 3457, 4057, 4705, 5401, 6145, 6937, 7777, 8665, 9601, 10585, 11617, 12697, 13825, 15001, 16225, 17497, 18817, 20185, 21601, 23065, 24577, 26137, 27745, 29401, 31105, 32857, 34657
OFFSET
0,2
COMMENTS
The identity (24*n^2 + 1)^2 - (144*n^2 + 12) * (2*n)^2 = 1 can be written as a(n)^2 - A158546(n) * A005843(n)^2 = 1.
FORMULA
G.f.: (1 + 22*x + 25*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/2 + coth(Pi/(2*sqrt(6))*Pi/(4*sqrt(6)).
Sum_{n>=0} (-1)^n/a(n) = 1/2 + cosech(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)). (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 25, 97}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
PROG
(Magma) I:=[1, 25, 97]; [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 14 2012
(PARI) for(n=0, 40, print1(24*n^2+1", ")); \\ Vincenzo Librandi, Feb 14 2012
CROSSREFS
Cf. A008594.
Sequence in context: A063769 A099771 A266818 * A144854 A237202 A353152
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 21 2009
EXTENSIONS
Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009
STATUS
approved