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A153169
a(n) = 4*n^2 + 12*n + 3.
1
19, 43, 75, 115, 163, 219, 283, 355, 435, 523, 619, 723, 835, 955, 1083, 1219, 1363, 1515, 1675, 1843, 2019, 2203, 2395, 2595, 2803, 3019, 3243, 3475, 3715, 3963, 4219, 4483, 4755, 5035
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
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
Sequence in context: A094845 A220446 A155770 * A090797 A078561 A074222
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Dec 20 2008
EXTENSIONS
Definition rewritten by Bruno Berselli, Jan 25 2012
STATUS
approved