login
A158543
a(n) = 144*n^2 - 12.
2
132, 564, 1284, 2292, 3588, 5172, 7044, 9204, 11652, 14388, 17412, 20724, 24324, 28212, 32388, 36852, 41604, 46644, 51972, 57588, 63492, 69684, 76164, 82932, 89988, 97332, 104964, 112884, 121092, 129588, 138372, 147444, 156804, 166452, 176388, 186612, 197124
OFFSET
1,1
COMMENTS
The identity (24*n^2 - 1)^2 - (144*n^2 - 12)*(2*n)^2 = 1 can be written as A158544(n)^2 - a(n)*A005843(n)^2 = 1.
FORMULA
From Vincenzo Librandi, Feb 14 2012: (Start)
G.f.: -x*(132 + 168*x - 12*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)))/24.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) - 1)/24. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {132, 564, 1284}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
144*Range[40]^2-12 (* Harvey P. Dale, Oct 20 2012 *)
PROG
(Magma) I:=[132, 564, 1284]; [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(144*n^2 - 12", ")); \\ Vincenzo Librandi, Feb 14 2012
CROSSREFS
Sequence in context: A204834 A241288 A115132 * A156958 A305271 A090199
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 21 2009
EXTENSIONS
Comment rewritten by R. J. Mathar, Oct 16 2009
STATUS
approved