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A045945 Hexagonal matchstick numbers: a(n) = 3*n(3*n+1). 8

%I

%S 0,12,42,90,156,240,342,462,600,756,930,1122,1332,1560,1806,2070,2352,

%T 2652,2970,3306,3660,4032,4422,4830,5256,5700,6162,6642,7140,7656,

%U 8190,8742,9312,9900,10506,11130,11772,12432,13110,13806,14520,15252,16002

%N Hexagonal matchstick numbers: a(n) = 3*n(3*n+1).

%C This may also be construed as the number of line segments illustrating the isometric projection of a cube of side length n. Moreover, a(n) equals the number of rods making a cube of side length n+1 minus the number of rods making a cube of side length n. See the illustration in the links and formula below.

%H Ivan Panchenko, <a href="/A045945/b045945.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter M. Chema, <a href="/A045945/a045945.pdf">Illustration of initial terms as the first difference of number of rods required to make a 3-D cube.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 6*(3*n-1) (with a(0)=0). - _Vincenzo Librandi_, Nov 18 2010

%F G.f.: 6*x*(2+x)/(1-x)^3. - _Colin Barker_, Feb 12 2012

%F a(n) = 6*A005449(n). - _R. J. Mathar_, Feb 13 2016

%F a(n) = A059986(n) - A059986(n-1). - _Peter M. Chema_, Feb 26 2017

%F a(n) = 6*(A000217(n) + A000290(n)). - _Peter M. Chema_, Mar 26 2017

%p a:= n-> 3*n*(3*n+1): seq(a(n), n=0..42); # _Zerinvary Lajos_, May 03 2007

%t f[n_]:=3*n*(3*n+1);f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011 *)

%o (PARI) for(n=0,50,print1(3*n*(3*n+1),", ")) \\ _Derek Orr_, Feb 26 2017

%o (PARI) a(n)=3*n*(3*n+1) \\ _Charles R Greathouse IV_, Feb 27 2017

%o (Python) def a(n): return 3*n*(3*n + 1) # _Indranil Ghosh_, Mar 26 2017

%Y Cf. A033580, A045946, A059986.

%K nonn,easy

%O 0,2

%A _R. K. Guy_

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Last modified October 17 07:05 EDT 2018. Contains 316276 sequences. (Running on oeis4.)