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a(n) = (6*n+3)^2.
15

%I #40 Aug 19 2022 13:51:38

%S 9,81,225,441,729,1089,1521,2025,2601,3249,3969,4761,5625,6561,7569,

%T 8649,9801,11025,12321,13689,15129,16641,18225,19881,21609,23409,

%U 25281,27225,29241,31329,33489,35721,38025,40401,42849,45369,47961,50625,53361,56169

%N a(n) = (6*n+3)^2.

%H Vincenzo Librandi, <a href="/A016946/b016946.txt">Table of n, a(n) for n = 0..2000</a>

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

%F a(n) = 36*A002378(n)+9. - _Jean-Bernard François_, Oct 12 2014

%F From _Wesley Ivan Hurt_, Oct 13 2014: (Start)

%F G.f.: 9*(1+6*x+x^2)/(1-x)^3.

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).

%F a(n) = A016945(n)^2 = A000290(A016945(n)). (End)

%F Sum_{n>=0} 1/a(n) = A086729. - _Amiram Eldar_, Nov 16 2020

%F a(n) = 9*A016754(n). - _R. J. Mathar_, Dec 11 2020

%F Sum_{n>=0} (-1)^n/a(n) = G/9, where G is Catalan's constant (A006752). - _Amiram Eldar_, Mar 30 2022

%F E.g.f.: 9*exp(x)*(1 + 8*x + 4*x^2). - _Stefano Spezia_, Aug 19 2022

%p A016946:=n->(6*n+3)^2: seq(A016946(n), n=0..50); # _Wesley Ivan Hurt_, Oct 13 2014

%t (6 Range[0, 50] + 3)^2 (* or *)

%t CoefficientList[Series[9 (1 + 6 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Oct 13 2014 *)

%t LinearRecurrence[{3,-3,1},{9,81,225},40] (* _Harvey P. Dale_, Jul 13 2015 *)

%o (Magma) [(6*n+3)^2: n in [0..60]]; // _Vincenzo Librandi_, May 05 2011

%o (PARI) a(n)=(6*n+3)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000290, A002378, A006752, A016754, A016945, A086729.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_.