A158463 a(n) = 12*n^2 - 1.

-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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Cf. A001082, A135453.

Sequence in context: A009941 A226155 A107149 * A143830 A339504 A178572

Adjacent sequences: A158460 A158461 A158462 * A158464 A158465 A158466

Keyword

sign,easy,changed

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