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A158659
a(n) = 784*n^2 + 28.
2
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
0,1
COMMENTS
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.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
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)
MATHEMATICA
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 *)
PROG
(Magma) 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
(PARI) for(n=0, 40, print1(784*n^2 + 28", ")); \\ Vincenzo Librandi, Feb 17 2012
CROSSREFS
Sequence in context: A114037 A041365 A042514 * A214133 A119055 A324450
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
EXTENSIONS
Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
STATUS
approved