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A158771
a(n) = 78*n^2 - 1.
2
77, 311, 701, 1247, 1949, 2807, 3821, 4991, 6317, 7799, 9437, 11231, 13181, 15287, 17549, 19967, 22541, 25271, 28157, 31199, 34397, 37751, 41261, 44927, 48749, 52727, 56861, 61151, 65597, 70199, 74957, 79871, 84941, 90167, 95549, 101087, 106781, 112631, 118637
OFFSET
1,1
COMMENTS
The identity (78*n^2 - 1)^2 - (1521*n^2 - 39)*(2*n)^2 = 1 can be written as a(n)^2 - A158770(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: x*(-77 - 80*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 23 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(78))*Pi/sqrt(78))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(78))*Pi/sqrt(78) - 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {77, 311, 701}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
78*Range[40]^2-1 (* Harvey P. Dale, Nov 28 2018 *)
PROG
(Magma) I:=[77, 311, 701]; [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 21 2012
(PARI) for(n=1, 40, print1(78*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 21 2012
CROSSREFS
Sequence in context: A156652 A298102 A158767 * A331976 A020206 A020304
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 26 2009
EXTENSIONS
Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009
STATUS
approved