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%I #153 Aug 19 2022 13:51:20
%S 0,6,24,54,96,150,216,294,384,486,600,726,864,1014,1176,1350,1536,
%T 1734,1944,2166,2400,2646,2904,3174,3456,3750,4056,4374,4704,5046,
%U 5400,5766,6144,6534,6936,7350,7776,8214,8664,9126,9600,10086,10584,11094,11616
%N a(n) = 6*n^2.
%C Number of edges of a complete 4-partite graph of order 4n, K_n,n,n,n. - _Roberto E. Martinez II_, Oct 18 2001
%C Number of edges of the complete bipartite graph of order 7n, K_n, 6n. - _Roberto E. Martinez II_, Jan 07 2002
%C 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
%C 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
%C a(n) can represented as n concentric hexagons (see example). - _Omar E. Pol_, Aug 21 2011
%C 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
%C 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
%C a(n+1) gives the number of vertices in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like 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
%C a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n). - _Sergey Pavlov_, Mar 31 2017
%C 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 k-gonal numbers. In this case k = 5. - _Omar E. Pol_, May 13 2018
%C The sequence also gives the number of size=1 triangles within a match-made hexagon of size n. - _John King_, Mar 31 2019
%C 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
%H Nathaniel Johnston, <a href="/A033581/b033581.txt">Table of n, a(n) for n = 0..10000</a>
%H Bruno Berselli, <a href="/A033581/a033581.jpg">An interpretation of initial terms</a>.
%H Ivan N. Ianakiev, <a href="http://mislandia.weebly.com/blog/hexagon-like-honeycomb-built-from-regular-congruent-hexagons">Hexagon-like honeycomb built form regular congruent hexagons</a>.
%H Leo Tavares, <a href="/A033581/a033581_1.jpg">Illustration: Diamond Star Rays</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlatonicSolid.html">Platonic Solid</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A000290(n)*6. - _Omar E. Pol_, Dec 11 2008
%F a(n) = A001105(n)*3 = A033428(n)*2. - _Omar E. Pol_, Dec 13 2008
%F a(n) = 12*n + a(n-1) - 6, with a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F G.f.: 6*x*(1+x)/(1-x)^3. - _Colin Barker_, Feb 14 2012
%F For n > 0: a(n) = A005897(n) - 2. - _Reinhard Zumkeller_, Apr 27 2014
%F a(n) = 3*floor(1/(1-cos(1/n))) = floor(1/(1-n*sin(1/n))) for n > 0. - _Clark Kimberling_, Oct 08 2014
%F 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
%F From _Amiram Eldar_, Feb 03 2021: (Start)
%F Sum_{n>=1} 1/a(n) = Pi^2/36.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 (A086729).
%F Product_{n>=1} (1 + 1/a(n)) = sqrt(6)*sinh(Pi/sqrt(6))/Pi.
%F Product_{n>=1} (1 - 1/a(n)) = sqrt(6)*sin(Pi/sqrt(6))/Pi. (End)
%F E.g.f.: 6*exp(x)*x*(1 + x). - _Stefano Spezia_, Aug 19 2022
%e From _Omar E. Pol_, Aug 21 2011: (Start)
%e Illustration of initial terms as concentric hexagons:
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%e . 6 24 54
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%e (End)
%p seq(6*n^2,n=0..44); # _Nathaniel Johnston_, Jun 26 2011
%t 6 Range[44]^2 (* _Michael De Vlieger_, Apr 02 2017 *)
%t LinearRecurrence[{3,-3,1},{0,6,24},50] (* _Harvey P. Dale_, Jul 03 2017 *)
%o (Haskell)
%o a033581 = (* 6) . (^ 2) -- _Reinhard Zumkeller_, Apr 27 2014
%o (PARI) vector(100,n,6*(n-1)^2) \\ _Derek Orr_, Mar 11 2015
%Y Bisection of A032528. Central column of triangle A001283.
%Y Cf. A000217, A000290, A001105, A009111, A033428, A033583, A085250, A086729, A152734, A152751.
%Y Cf. A000578, A001318, A003215, A005897, A045943, A045945, A045949, A070169, A071399.
%Y Cf. A017593 (first differences).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001
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