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%I #191 May 01 2023 10:03:27
%S 0,3,12,27,48,75,108,147,192,243,300,363,432,507,588,675,768,867,972,
%T 1083,1200,1323,1452,1587,1728,1875,2028,2187,2352,2523,2700,2883,
%U 3072,3267,3468,3675,3888,4107,4332,4563,4800,5043,5292,5547,5808,6075,6348
%N a(n) = 3*n^2.
%C The number of edges of a complete tripartite graph of order 3n, K_n,n,n. - _Roberto E. Martinez II_, Oct 18 2001
%C From _Floor van Lamoen_, Jul 21 2001: (Start)
%C Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,3,.... The spiral begins:
%C .
%C 33--32--31--30
%C / \
%C 34 16--15--14 29
%C / / \ \
%C 35 17 5---4 13 28
%C / / / \ \ \
%C 36 18 6 0---3--12--27--48-->
%C / / / / / / / /
%C 37 19 7 1---2 11 26 47
%C \ \ \ / / /
%C 38 20 8---9--10 25 46
%C \ \ / /
%C 39 21--22--23--24 45
%C \ /
%C 40--41--42--43--44
%C (End)
%C Number of edges of the complete bipartite graph of order 4n, K_n,3n. - _Roberto E. Martinez II_, Jan 07 2002
%C Also the number of partitions of 6n + 3 into at most 3 parts. - _R. K. Guy_, Oct 23, 2003
%C Also the number of partitions of 6n into exactly 3 parts. - _Colin Barker_, Mar 23 2015
%C Numbers n such that the imaginary quadratic field Q[sqrt(-n)] has six units. - _Marc LeBrun_, Apr 12 2006
%C The denominators of Hoehn's sequence (recalled by G. L. Honaker, Jr.) and the numerators of that sequence reversed. The sequence is 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11), (1+3+5+7)/(9+11+13+15), ...; reduced to 1/3, 4/12, 9/27, 16/48, ... . For the reversal, the reduction is 3/1, 12/4, 27/9, 48/16, ... . - _Enoch Haga_, Oct 05 2007
%C Right edge of tables in A200737 and A200741: A200737(n, A000292(n)) = A200741(n, A100440(n)) = a(n). - _Reinhard Zumkeller_, Nov 21 2011
%C The Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1<=i, j<=n, i/=j} (= the complete bipartite graph K(n,n) with horizontal edges removed). Example: a(3)=27 because G(3) is the cycle C(6) and 6*1 + 6*2 + 3*3 = 27. The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - _Emeric Deutsch_, Aug 29 2013
%C From _Michel Lagneau_, May 04 2015: (Start)
%C Integer area A of equilateral triangles whose side lengths are in the commutative ring Z[3^(1/4)] = {a + b*3^(1/4) + c*3^(1/2) + d*3^(3/4), a,b,c and d in Z}.
%C The area of an equilateral triangle of side length k is given by A = k^2*sqrt(3)/4. In the ring Z[3^(1/4)], if k = q*3^(1/4), then A = 3*q^2/4 is an integer if q is even. Example: 27 is in the sequence because the area of the triangle (6*3^(1/4), 6*3^(1/4), 6*3^(1/4)) is 27. (End)
%C a(n) is 2*sqrt(3) times the area of a 30-60-90 triangle with short side n. Also, 3 times the area of an n X n square. - _Wesley Ivan Hurt_, Apr 06 2016
%C Consider the hexagonal tiling of the plane. Extract any four hexagons adjacent by edge. This can be done in three ways. Fold the four hexagons so that all opposite faces occupy parallel planes. For all parallel projections of the resulting object, at least two correspond to area a(n) for side length of n of the original hexagons. - _Torlach Rush_, Aug 17 2022
%H Nathaniel Johnston, <a href="/A033428/b033428.txt">Table of n, a(n) for n = 0..10000</a>
%H Francesco Brenti and Paolo Sentinelli, <a href="https://arxiv.org/abs/2212.04932">Wachs permutations, Bruhat order and weak order</a>, arXiv:2212.04932 [math.CO], 2022.
%H A. J. C. Cunningham, <a href="/A056107/a056107.pdf">Factorisation of N and N' = (x^n -+ y^n) / (x -+ y [when x-y=n]</a>, Messenger Math., 54 (1924), 17-21. [Incomplete annotated scanned copy]
%H Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H Leo Tavares, <a href="/A033428/a033428_1.jpg">Hexagonal illustration</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Unit.html">Unit</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
%F G.f.: 3*x*(1+x)/(1-x)^3. - _R. J. Mathar_, Sep 09 2008
%F Main diagonal of triangle in A132111: a(n) = A132111(n,n). - _Reinhard Zumkeller_, Aug 10 2007
%F A214295(a(n)) = -1. - _Reinhard Zumkeller_, Jul 12 2012
%F a(n) = A215631(n,n) for n > 0. - _Reinhard Zumkeller_, Nov 11 2012
%F a(n) = A174709(6n+2). - _Philippe Deléham_, Mar 26 2013
%F a(n) = a(n-1) + 6*n - 3, with a(0)=0. - _Jean-Bernard François_, Oct 04 2013
%F E.g.f.: 3*x*(1 + x)*exp(x). - _Ilya Gutkovskiy_, Apr 13 2016
%F a(n) = t(3*n) - 3*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): A000217(3*n) - 3*A000217(n). - _Bruno Berselli_, Aug 31 2017
%F a(n) = A000326(n) + A005449(n). - _Bruce J. Nicholson_, Jan 10 2020
%F From _Amiram Eldar_, Jul 03 2020: (Start)
%F Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36. (End)
%F From _Amiram Eldar_, Feb 03 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = sqrt(3)*sinh(Pi/sqrt(3))/Pi.
%F Product_{n>=1} (1 - 1/a(n)) = sqrt(3)*sin(Pi/sqrt(3))/Pi. (End)
%F a(n) = A003215(n) - A016777(n). - _Leo Tavares_, Apr 29 2023
%e From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
%e Illustration of initial terms:
%e . o
%e . o o
%e . o o
%e . o o o o
%e . o o o o o o
%e . o o o o o o
%e . o o o o o o o o o
%e . o o o o o o o o o o o o
%e . o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o o o o o o
%e . n=1 n=2 n=3 n=4
%e (End)
%p seq(3*n^2, n=0..46); # _Nathaniel Johnston_, Jun 26 2011
%t 3 Range[0, 50]^2
%t LinearRecurrence[{3, -3, 1}, {0, 3, 12}, 50] (* _Harvey P. Dale_, Feb 16 2013 *)
%o (PARI) a(n)=3*n^2
%o (Maxima) makelist(3*n^2,n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o (Haskell)
%o a033428 = (* 3) . (^ 2)
%o a033428_list = 0 : 3 : 12 : zipWith (+) a033428_list
%o (map (* 3) $ tail $ zipWith (-) (tail a033428_list) a033428_list)
%o -- _Reinhard Zumkeller_, Jul 11 2013
%o (Magma) [3*n^2: n in [0..50]]; // _Vincenzo Librandi_, May 18 2015
%o (Python) def a(n): return 3 * (n**2) # _Torlach Rush_, Aug 25 2022
%Y Cf. A000567, A000217, A000290, A033581, A033583, A092205, A092206.
%Y 3 times n-gonal numbers: A045943, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
%Y Cf. A219056.
%Y Cf. A000326, A005449, A086463.
%Y Cf. A003215, A016777.
%K nonn,easy
%O 0,2
%A _Jeff Burch_
%E Better description from _N. J. A. Sloane_, May 15 1998
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