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%I #73 Mar 11 2022 12:57:04
%S 0,9,36,81,144,225,324,441,576,729,900,1089,1296,1521,1764,2025,2304,
%T 2601,2916,3249,3600,3969,4356,4761,5184,5625,6084,6561,7056,7569,
%U 8100,8649,9216,9801,10404,11025,11664,12321,12996,13689,14400,15129,15876
%N a(n) = (3*n)^2.
%C Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - _Roberto E. Martinez II_, Jan 07 2002
%C Area of a square with side 3n. - _Wesley Ivan Hurt_, Sep 24 2014
%C Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - _Peter Bala_, Jan 12 2022
%H Ivan Panchenko, <a href="/A016766/b016766.txt">Table of n, a(n) for n = 0..200</a>
%H John Elias, <a href="/A016766/a016766.png">Illustration of initial terms</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 9*n^2 = 9 * A000290(n). - _Omar E. Pol_, Dec 11 2008
%F a(n) = 3 * A033428(n). - _Omar E. Pol_, Dec 13 2008
%F a(n) = a(n-1) + 9*(2*n-1) for n>0, a(0)=0. - _Vincenzo Librandi_, Nov 19 2010
%F From _Wesley Ivan Hurt_, Sep 24 2014: (Start)
%F G.f.: 9*x*(1 + x)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>3.
%F a(n) = A000290(A008585(n)). (End)
%F From _Amiram Eldar_, Jan 25 2021: (Start)
%F Sum_{n>=1} 1/a(n) = Pi^2/54.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
%F Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
%F a(n) = A051624(n) + 8*A000217(n). In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - _Charlie Marion_, Mar 09 2022
%p A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # _Wesley Ivan Hurt_, Sep 24 2014
%t (3Range[0, 49])^2 (* _Alonso del Arte_, Sep 24 2014 *)
%o (Maxima) A016766(n):=(3*n)^2$
%o makelist(A016766(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [(3*n)^2 : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 24 2014
%o (PARI) a(n)=9*n^2 \\ _Charles R Greathouse IV_, Sep 28 2015
%Y Numbers of the form 9n^2 + kn, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - _Jason Kimberley_, Nov 09 2012
%Y Cf. A086089.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Zerinvary Lajos_, May 30 2006
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