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 A033428 a(n) = 3*n^2. 100
 0, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, 507, 588, 675, 768, 867, 972, 1083, 1200, 1323, 1452, 1587, 1728, 1875, 2028, 2187, 2352, 2523, 2700, 2883, 3072, 3267, 3468, 3675, 3888, 4107, 4332, 4563, 4800, 5043, 5292, 5547, 5808, 6075, 6348 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The number of edges of a complete tripartite graph of order 3n, K_n,n,n. - Roberto E. Martinez II, Oct 18 2001 From Floor van Lamoen, Jul 21 2001: (Start) 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: . 33--32--31--30 / \ 34 16--15--14 29 / / \ \ 35 17 5---4 13 28 / / / \ \ \ 36 18 6 0---3--12--27--48--> / / / / / / / / 37 19 7 1---2 11 26 47 \ \ \ / / / 38 20 8---9--10 25 46 \ \ / / 39 21--22--23--24 45 \ / 40--41--42--43--44 (End) Number of edges of the complete bipartite graph of order 4n, K_n,3n. - Roberto E. Martinez II, Jan 07 2002 Also the number of partitions of 6n + 3 into at most 3 parts. - R. K. Guy, Oct 23, 2003 Also the number of partitions of 6n into exactly 3 parts. - Colin Barker, Mar 23 2015 Numbers n such that the imaginary quadratic field Q[sqrt(-n)] has six units. - Marc LeBrun, Apr 12 2006 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 Right edge of tables in A200737 and A200741: A200737(n, A000292(n)) = A200741(n, A100440(n)) = a(n). - Reinhard Zumkeller, Nov 21 2011 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 From Michel Lagneau, May 04 2015: (Start) 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}. 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) 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 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 LINKS Nathaniel Johnston, Table of n, a(n) for n = 0..10000 A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y [when x-y=n], Messenger Math., 54 (1924), 17-21. [Incomplete annotated scanned copy] Frank Ellermann, Illustration of binomial transforms. Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. Eric Weisstein's World of Mathematics, Crown Graph. Eric Weisstein's World of Mathematics, Wiener Index. Eric Weisstein's World of Mathematics, Unit. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2. G.f.: 3*x*(1+x)/(1-x)^3. - R. J. Mathar, Sep 09 2008 Main diagonal of triangle in A132111: a(n) = A132111(n,n). - Reinhard Zumkeller, Aug 10 2007 A214295(a(n)) = -1. - Reinhard Zumkeller, Jul 12 2012 a(n) = A215631(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012 a(n) = A174709(6n+2). - Philippe Deléham, Mar 26 2013 a(n) = a(n-1) + 6*n - 3, with a(0)=0. - Jean-Bernard François, Oct 04 2013 E.g.f.: 3*x*(1 + x)*exp(x). - Ilya Gutkovskiy, Apr 13 2016 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 a(n) = A000326(n) + A005449(n). - Bruce J. Nicholson, Jan 10 2020 From Amiram Eldar, Jul 03 2020: (Start) Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463). Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36. (End) From Amiram Eldar, Feb 03 2021: (Start) Product_{n>=1} (1 + 1/a(n)) = sqrt(3)*sinh(Pi/sqrt(3))/Pi. Product_{n>=1} (1 - 1/a(n)) = sqrt(3)*sin(Pi/sqrt(3))/Pi. (End) EXAMPLE From Ilya Gutkovskiy, Apr 13 2016: (Start) Illustration of initial terms: . o . o o . o o . o o o o . o o o o o o . o o o o o o . o o o o o o o o o . o o o o o o o o o o o o . o o o o o o o o o o o o . o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o o o . n=1 n=2 n=3 n=4 (End) MAPLE seq(3*n^2, n=0..46); # Nathaniel Johnston, Jun 26 2011 MATHEMATICA 3 Range[0, 50]^2 LinearRecurrence[{3, -3, 1}, {0, 3, 12}, 50] (* Harvey P. Dale, Feb 16 2013 *) PROG (PARI) a(n)=3*n^2 (Maxima) makelist(3*n^2, n, 0, 30); /* Martin Ettl, Nov 12 2012 */ (Haskell) a033428 = (* 3) . (^ 2) a033428_list = 0 : 3 : 12 : zipWith (+) a033428_list (map (* 3) \$ tail \$ zipWith (-) (tail a033428_list) a033428_list) -- Reinhard Zumkeller, Jul 11 2013 (Magma) [3*n^2: n in [0..50]]; // Vincenzo Librandi, May 18 2015 (Python) def a(n): return 3 * (n**2) # Torlach Rush, Aug 25 2022 CROSSREFS Cf. A000567, A000217, A000290, A033581, A033583, A092205, A092206. 3 times n-gonal numbers: A045943, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875. Cf. A219056. Cf. A000326, A005449, A086463. Sequence in context: A125614 A061936 A074630 * A213486 A018230 A058034 Adjacent sequences: A033425 A033426 A033427 * A033429 A033430 A033431 KEYWORD nonn,easy AUTHOR EXTENSIONS Better description from N. J. A. Sloane, May 15 1998 STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)