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A158463
a(n) = 12*n^2 - 1.
5
-1, 11, 47, 107, 191, 299, 431, 587, 767, 971, 1199, 1451, 1727, 2027, 2351, 2699, 3071, 3467, 3887, 4331, 4799, 5291, 5807, 6347, 6911, 7499, 8111, 8747, 9407, 10091, 10799, 11531, 12287, 13067, 13871, 14699, 15551, 16427, 17327, 18251, 19199
OFFSET
0,2
COMMENTS
Related to Legendre polynomials - see Mma line. - N. J. A. Sloane, Nov 17 2009
One notices that this sequence produces an inordinate number of semiprimes, perhaps better than mere chance for large values of n. - J. M. Bergot, Jun 30 2011
Sequence found by reading the line from -1, in the direction -1, 11,..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
FORMULA
a(-n) = a(n).
G.f.: ( 1-14*x-11*x^2 ) / (x-1)^3 . - R. J. Mathar, Aug 27 2011
a(n) = A135453(n) - 1. - Omar E. Pol, Jul 18 2012
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(12))*cot(Pi/sqrt(12)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(12))*csc(Pi/sqrt(12)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(12))*csc(Pi/sqrt(12)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(12))*sin(Pi/sqrt(6))/sqrt(2). (End)
From Gerry Martens, Apr 06 2024: (Start)
a(n) = Re((2*n*i-1)^3).
a(n) = -8*(1/4+n^2)^(3/2)*cos(3*arctan(2*n)). (End)
EXAMPLE
G.f. = -1 + 11*x + 47*x^2 + 107*x^3 + 191*x^4 + 299*x^5 + 431*x^6 + 587*x^7 + 767*x^8 + ...
MATHEMATICA
Table[Numerator[LegendreP[2, 2n]], {n, 0, 100}] - N. J. A. Sloane, Nov 17 2009
a[ n_] := 2 LegendreP[ 2, 2 n] (* Michael Somos, Jul 01 2011 *)
12*Range[0, 40]^2-1 (* or *) LinearRecurrence[{3, -3, 1}, {-1, 11, 47}, 50] (* Harvey P. Dale, Jun 22 2019 *)
PROG
(PARI) {a(n) = 12 * n^2 - 1} /* Michael Somos, Nov 12 2011 */
(Magma) [12*n^2 - 1: n in [0..100]]; // G. C. Greubel, Sep 25 2018
CROSSREFS
Sequence in context: A009941 A226155 A107149 * A143830 A339504 A178572
KEYWORD
sign,easy
AUTHOR
Vincenzo Librandi, Mar 19 2009
EXTENSIONS
Minor edits by N. J. A. Sloane, Nov 17 2009
Comment rewritten by Bruno Berselli, Aug 27 2011
STATUS
approved