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a(n) = 4*n^2 + 12*n + 3.
1

%I #26 Mar 02 2023 02:54:58

%S 19,43,75,115,163,219,283,355,435,523,619,723,835,955,1083,1219,1363,

%T 1515,1675,1843,2019,2203,2395,2595,2803,3019,3243,3475,3715,3963,

%U 4219,4483,4755,5035

%N a(n) = 4*n^2 + 12*n + 3.

%C 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.

%C The complete list of nonnegative values of x in x^3 + 6x^2 = y^2 is given by A028878. - _Bruno Berselli_, Jan 25 2012

%H Vincenzo Librandi, <a href="/A153169/b153169.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Colin Barker_, Jan 24 2012: (Start)

%F G.f.: x*(19 - 14*x + 3*x^2)/(1-x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=19, a(2)=43, a(3)=75. (End)

%F Sum_{n>=1} 1/a(n) = -2/15 + tan(sqrt(3/2)*Pi)*Pi/(4*sqrt(6)). - _Amiram Eldar_, Mar 02 2023

%t LinearRecurrence[{3, -3, 1}, {19, 43, 75}, 50] (* _Vincenzo Librandi_, Feb 25 2012 *)

%o (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

%o (PARI) for(n=1, 50, print1(4*n^2 + 12*n + 3", ")); \\ _Vincenzo Librandi_, Feb 25 2012

%Y Cf. A028878, A153167.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Dec 20 2008

%E Definition rewritten by _Bruno Berselli_, Jan 25 2012