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%I #52 Mar 09 2024 08:14:48
%S 0,12,54,144,300,540,882,1344,1944,2700,3630,4752,6084,7644,9450,
%T 11520,13872,16524,19494,22800,26460,30492,34914,39744,45000,50700,
%U 56862,63504,70644,78300,86490,95232,104544,114444,124950,136080,147852,160284,173394
%N Number of rods required to make a 3-D cube of side length n.
%C Equals number of rods making a cube of side length n+1 minus the number of line segments illustrating the isometric projection of a cube of side length n+1 (i.e., the hexagonal matchstick numbers). See the illustration in the links and formula below. - _Peter M. Chema_, Mar 14 2017
%C a(n) is also the edge count and intersection number of the (n+1) X (n+1) X (n+1) grid graph. - _Eric W. Weisstein_, Mar 09 2024
%H Peter M. Chema, <a href="/A059986/a059986.pdf">First difference are the hexagonal matchstick numbers or isometric projection of a cube</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntersectionNumber.html">Intersection Number</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 3*n*(n+1)^2. - Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005
%F From _Geoffrey Critzer_, May 17 2009: (Start)
%F a(n) = a(n-1) + 9*n^2 + 3*n.
%F O.g.f.: 6*x*(2 + x)/(1 - x)^4.
%F E.g.f.: 3*x*exp(x)*(x^2 + 5*x + 4). (End)
%F a(n) = A117227(n^3). - _Michel Marcus_, Jun 19 2013
%F For n > 0, a(n) = Sum_{k=1..n} 2*(n+1)(k+n+1), which is the sum of all perimeters of Pythagorean triangles with arms 2*k*(n+1) and (n+1)^2 - k^2 with hypotenuse k^2 + (n+1)^2. - _J. M. Bergot_, May 12 2014
%F a(n) = a(n+1) - A045945(n+1). - _Peter M. Chema_, Mar 14 2017
%F a(n) = (n-1)*t(n+1) + n*(t(n)+t(n+1)) + (n+1)*(t(n-1)+t(n)+t(n+1)), where t = A000217. - _J. M. Bergot_, May 30 2017
%F From _Amiram Eldar_, Jan 14 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 2/3 - Pi^2/18.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = -2/3 + Pi^2/36 + 2*log(2)/3. (End)
%e A 1 X 1 X 1 cube requires 12 rods.
%p A059986:=n->3*n*(n+1)^2; seq(A059986(n), n=0..50); # _Wesley Ivan Hurt_, May 13 2014
%t Table[EdgeCount[GridGraph[{n, n, n}]], {n, 39}] (* _Geoffrey Critzer_, May 17 2009 *)
%t Table[3 n (n + 1)^2, {n, 0, 50}] (* _Wesley Ivan Hurt_, May 13 2014 *)
%t LinearRecurrence[{4, -6, 4, -1}, {0, 12, 54, 144}, 20] (* _Eric W. Weisstein_, Mar 09 2024 *)
%t CoefficientList[Series[6 x (2 + x)/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 09 2024 *)
%o (Magma) [3*n*(n+1)^2: n in [0..50]]; // _Wesley Ivan Hurt_, May 13 2014
%o (PARI) a(n) = 3*n*(n+1)^2 \\ _Charles R Greathouse IV_, May 14 2014
%Y Cf. A045945, A117227.
%K nonn,easy
%O 0,2
%A Laura Twomey (sxe15(AT)hotmail.com), Mar 07 2001
%E More terms from Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005
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