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URL: https://oeis.org/A145678

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A145678
a(n) = 441*n^2 - 21.
2
420, 1743, 3948, 7035, 11004, 15855, 21588, 28203, 35700, 44079, 53340, 63483, 74508, 86415, 99204, 112875, 127428, 142863, 159180, 176379, 194460, 213423, 233268, 253995, 275604, 298095, 321468, 345723, 370860, 396879, 423780, 451563, 480228, 509775, 540204, 571515
OFFSET
1,1
COMMENTS
The identity (42*n^2 - 1)^2 - (441*n^2 - 21)*(2*n)^2 = 1 can be written as A158626(n)^2 - a(n)*A005843(n)^2 = 1.
FORMULA
From Vincenzo Librandi, Feb 12 2012: (Start)
G.f.: -21*x*(20 + 23*x - x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(21))*Pi/sqrt(21))/42.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(21))*Pi/sqrt(21) - 1)/42. (End)
E.g.f.: 21*(exp(x)*(21*x^2 + 21*x - 1) + 1). - Elmo R. Oliveira, Dec 26 2025
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {420, 1743, 3948}, 50] (* Vincenzo Librandi, Feb 12 2012 *)
PROG
(Magma) I:=[420, 1743, 3948]; [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 12 2012
(PARI) for(n=1, 40, print1(441*n^2-21", ")); \\ Vincenzo Librandi, Feb 12 2012
CROSSREFS
Sequence in context: A235233 A251084 A250383 * A160372 A171259 A061125
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
STATUS
approved