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A033428
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a(n) = 3*n^2.
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103
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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
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OFFSET
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0,2
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COMMENTS
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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,.... 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)
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
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
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
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LINKS
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Eric Weisstein's World of Mathematics, Unit.
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FORMULA
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a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36. (End)
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)
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EXAMPLE
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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)
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MAPLE
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MATHEMATICA
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3 Range[0, 50]^2
LinearRecurrence[{3, -3, 1}, {0, 3, 12}, 50] (* Harvey P. Dale, Feb 16 2013 *)
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PROG
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(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)
(Python) def a(n): return 3 * (n**2) # Torlach Rush, Aug 25 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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