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A158658
a(n) = 56*n^2 - 1.
2
55, 223, 503, 895, 1399, 2015, 2743, 3583, 4535, 5599, 6775, 8063, 9463, 10975, 12599, 14335, 16183, 18143, 20215, 22399, 24695, 27103, 29623, 32255, 34999, 37855, 40823, 43903, 47095, 50399, 53815, 57343, 60983, 64735, 68599, 72575, 76663, 80863, 85175, 89599
OFFSET
1,1
COMMENTS
The identity (56*n^2 - 1)^2 - (784*n^2 - 2 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158657(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: x*(-55 - 58*x + 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>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(14)))*Pi/(2*sqrt(14)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(14)))*Pi/(2*sqrt(14)) - 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {55, 223, 503}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
56 Range[40]^2-1 (* Harvey P. Dale, Oct 23 2022 *)
PROG
(Magma) I:=[55, 223, 503]; [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=1, 40, print1(56*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 17 2012
CROSSREFS
Sequence in context: A254148 A271738 A280888 * A296036 A286854 A020182
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