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A158540
a(n) = 22*n^2 - 1.
2
21, 87, 197, 351, 549, 791, 1077, 1407, 1781, 2199, 2661, 3167, 3717, 4311, 4949, 5631, 6357, 7127, 7941, 8799, 9701, 10647, 11637, 12671, 13749, 14871, 16037, 17247, 18501, 19799, 21141, 22527, 23957, 25431, 26949, 28511, 30117, 31767, 33461, 35199, 36981, 38807
OFFSET
1,1
COMMENTS
The identity (22*n^2 - 1)^2 - (121*n^2 - 11)*(2*n)^2 = 1 can be written as a(n)^2 - A158539(n)*A005843(n)^2 = 1.
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(-21 - 24*x + x^2)/(x-1)^3.
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(22))*Pi/sqrt(22))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(22))*Pi/sqrt(22) - 1)/2. (End)
MATHEMATICA
22Range[40]^2-1 (* or *) LinearRecurrence[{3, -3, 1}, {21, 87, 197}, 40]
PROG
(Magma) I:=[21, 87, 197]; [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=1, 40, print1(22*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 14 2012
CROSSREFS
Sequence in context: A301540 A240518 A219850 * A304514 A219886 A211464
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 21 2009
EXTENSIONS
Comment rewritten by R. J. Mathar, Oct 16 2009
STATUS
approved