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URL: https://oeis.org/A158686

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A158686
a(n) = 64*n^2 + 1.
4
1, 65, 257, 577, 1025, 1601, 2305, 3137, 4097, 5185, 6401, 7745, 9217, 10817, 12545, 14401, 16385, 18497, 20737, 23105, 25601, 28225, 30977, 33857, 36865, 40001, 43265, 46657, 50177, 53825, 57601, 61505, 65537, 69697, 73985, 78401, 82945, 87617, 92417, 97345, 102401
OFFSET
0,2
COMMENTS
The identity (64*n^2 + 1)^2 - (1024*n^2 + 32)*(2*n)^2 = 1 can be written as a(n)^2 - A158685(n)*(A005843(n))^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: -(1+62*x+65*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 21 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/8)*Pi/8 + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/8)*Pi/8 + 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(1 + 64*x + 64*x^2).
a(n) = A108211(2*n) for n > 0. (End)
MATHEMATICA
64 Range[0, 40]^2 + 1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 65, 257}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[- (1 + 62 x + 65 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)
PROG
(Magma) [64*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013
(PARI) a(n)=64*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 24 2009
EXTENSIONS
Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009
STATUS
approved