cp's OEIS Frontend

A016766 a(n) = (3*n)^2.

%I A016766 #78 Nov 30 2024 17:02:36
%S A016766 0,9,36,81,144,225,324,441,576,729,900,1089,1296,1521,1764,2025,2304,
%T A016766 2601,2916,3249,3600,3969,4356,4761,5184,5625,6084,6561,7056,7569,
%U A016766 8100,8649,9216,9801,10404,11025,11664,12321,12996,13689,14400,15129,15876,16641,17424
%N A016766 a(n) = (3*n)^2.
%C A016766 Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - _Roberto E. Martinez II_, Jan 07 2002
%C A016766 Area of a square with side 3n. - _Wesley Ivan Hurt_, Sep 24 2014
%C A016766 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 A016766 Ivan Panchenko, <a href="/A016766/b016766.txt">Table of n, a(n) for n = 0..200</a>
%H A016766 John Elias, <a href="/A016766/a016766.png">Illustration of initial terms</a>.
%H A016766 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A016766 a(n) = 9*n^2 = 9*A000290(n). - _Omar E. Pol_, Dec 11 2008
%F A016766 a(n) = 3*A033428(n). - _Omar E. Pol_, Dec 13 2008
%F A016766 a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - _Vincenzo Librandi_, Nov 19 2010
%F A016766 From _Wesley Ivan Hurt_, Sep 24 2014: (Start)
%F A016766 G.f.: 9*x*(1 + x)/(1 - x)^3.
%F A016766 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
%F A016766 a(n) = A000290(A008585(n)). (End)
%F A016766 From _Amiram Eldar_, Jan 25 2021: (Start)
%F A016766 Sum_{n>=1} 1/a(n) = Pi^2/54.
%F A016766 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
%F A016766 Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
%F A016766 Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
%F A016766 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
%F A016766 From _Elmo R. Oliveira_, Nov 30 2024: (Start)
%F A016766 E.g.f.: 9*x*(1 + x)*exp(x).
%F A016766 a(n) = n*A008591(n) = A195042(2*n). (End)
%p A016766 A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # _Wesley Ivan Hurt_, Sep 24 2014
%t A016766 (3Range[0, 49])^2 (* _Alonso del Arte_, Sep 24 2014 *)
%o A016766 (Maxima) A016766(n):=(3*n)^2$
%o A016766 makelist(A016766(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o A016766 (Magma) [(3*n)^2 : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 24 2014
%o A016766 (PARI) a(n)=9*n^2 \\ _Charles R Greathouse IV_, Sep 28 2015
%Y A016766 Numbers of the form 9*n^2 + k*n, 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 A016766 Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.
%K A016766 nonn,easy
%O A016766 0,2
%A A016766 _N. J. A. Sloane_
%E A016766 More terms from _Zerinvary Lajos_, May 30 2006