



0, 6, 24, 54, 96, 150, 216, 294, 384, 486, 600, 726, 864, 1014, 1176, 1350, 1536, 1734, 1944, 2166, 2400, 2646, 2904, 3174, 3456, 3750, 4056, 4374, 4704, 5046, 5400, 5766, 6144, 6534, 6936, 7350, 7776, 8214, 8664, 9126, 9600, 10086, 10584, 11094, 11616
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OFFSET

0,2


COMMENTS

Number of edges in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n).  Roberto E. Martinez II, Jan 07 2002
Total surface area of a cube of edge length n. See A000578 for cube volume. See A070169 and A071399 for surface area and volume of a regular tetrahedron and links for the other Platonic solids.  Rick L. Shepherd, Apr 24 2002
a(n) can represented as n concentric hexagons (see example).  Omar E. Pol, Aug 21 2011
Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A003154 in the same spiral.  Omar E. Pol, Sep 08 2011
Together with 1, numbers m such that floor(2*m/3) and floor(3*m/2) are both squares. Example: floor(2*150/3) = 100 and floor(3*150/2) = 225 are both squares, so 150 is in the sequence.  Bruno Berselli, Sep 15 2014
a(n+1) gives the number of vertices in a hexagonlike honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagonlike honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter has 18 vertices. The core hexagon has 6 vertices. a(2) = 18 + 6 = 24 is the total number of vertices.  Ivan N. Ianakiev, Mar 11 2015
a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n).  Sergey Pavlov, Mar 31 2017
More generally, if k >= 5 we have that the sequence whose formula is a(n) = (2*k  4)*n^2 is also the sequence found by reading the line from 0, in the direction 0, (2*k  4), ..., in the square spiral whose vertices are the generalized kgonal numbers. In this case k = 5.  Omar E. Pol, May 13 2018
The sequence also gives the number of size=1 triangles within a matchmade hexagon of size n.  John King, Mar 31 2019
For hexagons, the number of matches required is A045945; thus number of size=1 triangles is A033581; number of larger triangles is A307253 and total number of triangles is A045949. See A045943 for analogs for Triangles; see A045946 for analogs for Stars.  John King, Apr 04 2019


LINKS



FORMULA

a(n) = 3*floor(1/(1cos(1/n))) = floor(1/(1n*sin(1/n))) for n > 0.  Clark Kimberling, Oct 08 2014
a(n) = t(4*n)  4*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(4*n)  4*A000217(n).  Bruno Berselli, Aug 31 2017
Sum_{n>=1} 1/a(n) = Pi^2/36.
Sum_{n>=1} (1)^(n+1)/a(n) = Pi^2/72 (A086729).
Product_{n>=1} (1 + 1/a(n)) = sqrt(6)*sinh(Pi/sqrt(6))/Pi.
Product_{n>=1} (1  1/a(n)) = sqrt(6)*sin(Pi/sqrt(6))/Pi. (End)


EXAMPLE

Illustration of initial terms as concentric hexagons:
.
. 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
.
. 6 24 54
.
(End)


MAPLE



MATHEMATICA

LinearRecurrence[{3, 3, 1}, {0, 6, 24}, 50] (* Harvey P. Dale, Jul 03 2017 *)


PROG

(Haskell)
(PARI) vector(100, n, 6*(n1)^2) \\ Derek Orr, Mar 11 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001


STATUS

approved



