A158556 -id:A158556 - cp's OEIS frontend

A158555 a(n) = 196*n^2 + 14.

14, 210, 798, 1778, 3150, 4914, 7070, 9618, 12558, 15890, 19614, 23730, 28238, 33138, 38430, 44114, 50190, 56658, 63518, 70770, 78414, 86450, 94878, 103698, 112910, 122514, 132510, 142898, 153678, 164850, 176414, 188370, 200718, 213458, 226590, 240114, 254030

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (28*n^2 + 1)^2 -(196*n^2 + 14)*(2*n)^2 = 1 can be written as A158556(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158556.

I:=[14, 210, 798]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 14 2012

LinearRecurrence[{3, -3, 1}, {14, 210, 798}, 50] (* Vincenzo Librandi, Feb 05 2012 *)

for(n=0, 40, print1(196*n^2 + 14", ")); \\ Vincenzo Librandi, Feb 14 2012

G.f.: 14*(1 + 12*x + 15*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(14))*Pi/sqrt(14) + 1)/28.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(14))*Pi/sqrt(14) + 1)/28. (End)
E.g.f.: 14*exp(x)*(1 + 14*x + 14*x^2). - Elmo R. Oliveira, Jan 15 2025

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

A158659 a(n) = 784*n^2 + 28.

Original entry on oeis.org

28, 812, 3164, 7084, 12572, 19628, 28252, 38444, 50204, 63532, 78428, 94892, 112924, 132524, 153692, 176428, 200732, 226604, 254044, 283052, 313628, 345772, 379484, 414764, 451612, 490028, 530012, 571564, 614684, 659372, 705628, 753452, 802844, 853804, 906332

Offset: 0

Vincenzo Librandi, Mar 23 2009

The identity (56*n^2 + 1)^2 - (784*n^2 + 28)*(2*n)^2 = 1 can be written as A158660(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..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, A158556, A158660.

I:=[28, 812, 3164]; [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}, {28, 812, 3164}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
784 Range[0,40]^2+28 (* Harvey P. Dale, Nov 01 2024 *)

for(n=0, 40, print1(784*n^2 + 28", ")); \\ Vincenzo Librandi, Feb 17 2012

G.f.: -28*(1 + 26*x + 29*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 20 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)) + 1)/56.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)) + 1)/56. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 28*exp(x)*(1 + 28*x + 28*x^2).
a(n) = 28*A158556(n). (End)

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

cp's OEIS Frontend

A158555 a(n) = 196*n^2 + 14.

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