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A033428 a(n) = 3*n^2. 83
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

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,... - Floor van Lamoen, Jul 21 2001. The spiral begins:

......16..15..14

....17..5...4...13

..18..6...0...3...12

19..7...1...2...11..26

..20..8...9...10..25

....21..22..23..24

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

A214295(a(n)) = -1. - Reinhard Zumkeller, Jul 12 2012

a(n) = A215631(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012

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

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]

F. 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, Unit

Eric Weisstein's World of Mathematics, Crown Graph.

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

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

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

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.

Sequence in context: A125614 A061936 A074630 * A213486 A018230 A058034

Adjacent sequences:  A033425 A033426 A033427 * A033429 A033430 A033431

KEYWORD

nonn,easy

AUTHOR

Jeff Burch

EXTENSIONS

Better description from N. J. A. Sloane, May 15 1998

STATUS

approved

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Last modified March 27 06:40 EDT 2017. Contains 284144 sequences.