|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
a(-n) = a(n).
G.f.: ( 1-14*x-11*x^2 ) / (x-1)^3 . - R. J. Mathar, Aug 27 2011
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)
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
12*Range[0, 40]^2-1 (* or *) LinearRecurrence[{3, -3, 1}, {-1, 11, 47}, 50] (* Harvey P. Dale, Jun 22 2019 *)
|
|
PROG
|
(Magma) [12*n^2 - 1: n in [0..100]]; // G. C. Greubel, Sep 25 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|