OFFSET
1,1
COMMENTS
Sequence gives values of x such that x^3 + 6x^2 = y^2 since a(n)^3 + 6*a(n)^2 = (8n^3 + 36n^2 + 42n + 9)^2.
The complete list of nonnegative values of x in x^3 + 6x^2 = y^2 is given by A028878. - Bruno Berselli, Jan 25 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Colin Barker, Jan 24 2012: (Start)
G.f.: x*(19 - 14*x + 3*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=19, a(2)=43, a(3)=75. (End)
Sum_{n>=1} 1/a(n) = -2/15 + tan(sqrt(3/2)*Pi)*Pi/(4*sqrt(6)). - Amiram Eldar, Mar 02 2023
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {19, 43, 75}, 50] (* Vincenzo Librandi, Feb 25 2012 *)
PROG
(Magma) I:=[19, 43, 75]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 25 2012
(PARI) for(n=1, 50, print1(4*n^2 + 12*n + 3", ")); \\ Vincenzo Librandi, Feb 25 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Dec 20 2008
EXTENSIONS
Definition rewritten by Bruno Berselli, Jan 25 2012
STATUS
approved