A158544 -id:A158544 - cp's OEIS frontend

A158543 a(n) = 144*n^2 - 12.

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, 207924

Offset: 1

Vincenzo Librandi, Mar 21 2009

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.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158463, A158544.

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

LinearRecurrence[{3, -3, 1}, {132, 564, 1284}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
144*Range[40]^2-12 (* Harvey P. Dale, Oct 20 2012 *)

for(n=1, 40, print1(144*n^2 - 12", ")); \\ Vincenzo Librandi, Feb 14 2012

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)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 12*(exp(x)*(12*x^2 + 12*x - 1) + 1).
a(n) = 12*A158463(n). (End)

Comment rewritten by R. J. Mathar, Oct 16 2009

A158636 a(n) = 576*n^2 - 24.

Original entry on oeis.org

552, 2280, 5160, 9192, 14376, 20712, 28200, 36840, 46632, 57576, 69672, 82920, 97320, 112872, 129576, 147432, 166440, 186600, 207912, 230376, 253992, 278760, 304680, 331752, 359976, 389352, 419880, 451560, 484392, 518376, 553512, 589800, 627240, 665832, 705576

Offset: 1

Vincenzo Librandi, Mar 23 2009

The identity (48*n^2 - 1)^2 - (576*n^2 - 24)*(2*n)^2 = 1 can be written as A065532(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A065532, A158544.

I:=[552, 2280, 5160]; [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 17 2012

LinearRecurrence[{3, -3, 1}, {552, 2280, 5160}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

for(n=1, 40, print1(576*n^2 - 24", ")); \\ Vincenzo Librandi, Feb 17 2012

G.f.: 24*x*(-23 - 26*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 19 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)))/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)) - 1)/48. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 24*(exp(x)*(24*x^2 + 24*x - 1) + 1).
a(n) = 24*A158544(n). (End)

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

cp's OEIS Frontend

A158543 a(n) = 144*n^2 - 12.

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