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%I #53 Feb 27 2022 05:46:34
%S 0,6,30,72,132,210,306,420,552,702,870,1056,1260,1482,1722,1980,2256,
%T 2550,2862,3192,3540,3906,4290,4692,5112,5550,6006,6480,6972,7482,
%U 8010,8556,9120,9702,10302,10920,11556,12210,12882,13572,14280,15006,15750,16512,17292
%N 6 times pentagonal numbers: a(n) = 3*n*(3*n-1).
%C a(n) is also the Wiener index of the windmill graph D(4,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3,n), D(5,n), and D(6,n) see A033991, A028994, and A180577, respectively. - _Emeric Deutsch_, Sep 21 2010
%C a(n+1) gives the number of edges 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 consists of 18 external edges. There are 6 edges shared by the perimeter hexagons. The core hexagon has 6 edges. a(2) is the total number of edges, i.e. 18 + 6 + 6 = 30. - _Ivan N. Ianakiev_, Mar 10 2015
%H Ivan Panchenko, <a href="/A152743/b152743.txt">Table of n, a(n) for n = 0..1000</a>
%H Ivan N. Ianakiev, <a href="http://mislandia.weebly.com/blog/hexagon-like-honeycomb-built-from-regular-congruent-hexagons">Hexagon-like honeycomb built from regular congruent hexagons</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WindmillGraph.html">Windmill Graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 9n^2 - 3n = A000326(n)*6.
%F a(n) = A049450(n)*3 = A062741(n)*2. - _Omar E. Pol_, Dec 15 2008
%F a(n) = a(n-1) + 18*n - 12 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010
%F G.f.: -((6*x*(2*x+1))/(x-1)^3). - _Harvey P. Dale_, Jun 30 2011
%F E.g.f.: 3*x*(2+3*x)*exp(x). - _G. C. Greubel_, Sep 01 2018
%F From _Amiram Eldar_, Feb 27 2022: (Start)
%F Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/18.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi*sqrt(3) - 6*log(2))/9. (End)
%p A152743:=n->3*n*(3*n-1); seq(A152743(n), n=0..50); # _Wesley Ivan Hurt_, Jun 09 2014
%t Table[3n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,6,30},40] (* _Harvey P. Dale_, Jun 30 2011 *)
%t CoefficientList[Series[-6x (2x+1)/(x-1)^3,{x,0,40}],x] (* _Robert G. Wilson v_, Mar 10 2015 *)
%o (Magma) [ 3*n*(3*n-1) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o (PARI) a(n)=3*n*(3*n-1) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000326, A152734, A152744, A049450, A062741, A033991, A028994, A180577.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, Dec 12 2008
%E Converted reference to link by _Omar E. Pol_, Oct 07 2010
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