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 A059986 Number of rods required to make a 3-D cube of side length n. 4
 0, 12, 54, 144, 300, 540, 882, 1344, 1944, 2700, 3630, 4752, 6084, 7644, 9450, 11520, 13872, 16524, 19494, 22800, 26460, 30492, 34914, 39744, 45000, 50700, 56862, 63504, 70644, 78300, 86490, 95232, 104544, 114444, 124950, 136080, 147852, 160284, 173394 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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 LINKS Table of n, a(n) for n=0..38. Peter M. Chema, First difference are the hexagonal matchstick numbers or isometric projection of a cube. Eric Weisstein's World of Mathematics, Edge Count. Eric Weisstein's World of Mathematics, Grid Graph. Eric Weisstein's World of Mathematics, Intersection Number. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n) = 3*n*(n+1)^2. - Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005 From Geoffrey Critzer, May 17 2009: (Start) a(n) = a(n-1) + 9*n^2 + 3*n. O.g.f.: 6*x*(2 + x)/(1 - x)^4. E.g.f.: 3*x*exp(x)*(x^2 + 5*x + 4). (End) a(n) = A117227(n^3). - Michel Marcus, Jun 19 2013 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 a(n) = a(n+1) - A045945(n+1). - Peter M. Chema, Mar 14 2017 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 From Amiram Eldar, Jan 14 2021: (Start) Sum_{n>=1} 1/a(n) = 2/3 - Pi^2/18. Sum_{n>=1} (-1)^(n+1)/a(n) = -2/3 + Pi^2/36 + 2*log(2)/3. (End) EXAMPLE A 1 X 1 X 1 cube requires 12 rods. MAPLE A059986:=n->3*n*(n+1)^2; seq(A059986(n), n=0..50); # Wesley Ivan Hurt, May 13 2014 MATHEMATICA Table[EdgeCount[GridGraph[{n, n, n}]], {n, 39}] (* Geoffrey Critzer, May 17 2009 *) Table[3 n (n + 1)^2, {n, 0, 50}] (* Wesley Ivan Hurt, May 13 2014 *) LinearRecurrence[{4, -6, 4, -1}, {0, 12, 54, 144}, 20] (* Eric W. Weisstein, Mar 09 2024 *) CoefficientList[Series[6 x (2 + x)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Mar 09 2024 *) PROG (Magma) [3*n*(n+1)^2: n in [0..50]]; // Wesley Ivan Hurt, May 13 2014 (PARI) a(n) = 3*n*(n+1)^2 \\ Charles R Greathouse IV, May 14 2014 CROSSREFS Cf. A045945, A117227. Sequence in context: A022704 A372025 A060785 * A088941 A019582 A025204 Adjacent sequences: A059983 A059984 A059985 * A059987 A059988 A059989 KEYWORD nonn,easy AUTHOR Laura Twomey (sxe15(AT)hotmail.com), Mar 07 2001 EXTENSIONS More terms from Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2005 STATUS approved

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Last modified September 14 12:31 EDT 2024. Contains 375921 sequences. (Running on oeis4.)