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Maximal number of bonds joining n nodes in simple cubic lattice.
8

%I #25 Aug 22 2021 19:15:10

%S 0,1,2,4,5,7,9,12,13,15,17,20,21,23,25,28,30,33,34,36,38,41,43,46,48,

%T 51,54,55,57,59,62,64,67,69,72,75,76,78,80,83,85,88,90,93,96,98,101,

%U 104,105,107,109,112,114,117,119,122,125,127,130,133,135,138,141

%N Maximal number of bonds joining n nodes in simple cubic lattice.

%C a(n) is also the maximal number of kisses between n cubes to form a polycube. The surface area of such polycubes are A193416. - _Mohammed Yaseen_, Aug 08 2021

%H Martin Y. Veillette, <a href="/A007818/b007818.txt">Table of n, a(n) for n = 1..500</a>

%H G. Agnarsson, <a href="http://arxiv.org/abs/1106.4997">On the number of hypercubic bipartitions of an integer</a>, arXiv preprint arXiv:1106.4997 [math.CO], 2011.

%H G. Agnarsson, <a href="http://arxiv.org/abs/1112.3015">Induced subgraphs of hypercubes</a>, arXiv preprint arXiv:1112.3015 [math.CO], 2011.

%H G. Agnarsson and K. Lauria, <a href="http://arxiv.org/abs/1302.6517">Extremal subgraphs of the d-dimensional grid graph</a>, arXiv preprint arXiv:1302.6517 [math.CO], 2013.

%F a(n) = 3*n - A193416(n)/2. - _Mohammed Yaseen_, Aug 08 2021

%t qmax = 2000; sequence =

%t FoldList[Plus, 0, q = Table[3, {qmax}];

%t q[[Flatten[

%t Table[Table[{j^3 + i (i - 1), j^3 + i^2, j^2 (j + 1) + i (i + 1),

%t j^2 (j + 1) + i^2, (j + 1)^2 j + i (i + 1), (j + 1)^2 j +

%t i^2}, {i, 1, j}], {j, 0, (qmax)^(1/3) - 1}]]]]--;

%t q[[Flatten[

%t Table[{j^3, j^2 (j + 1), (j + 1)^2 j}, {j,

%t 1, (qmax)^(1/3) - 1}]]]]--; q] (* _Martin Y. Veillette_, Jul 19 2011 *)

%Y Cf. A193416.

%K nonn

%O 1,3

%A D. Heuer (heuer(AT)isnd23.in2p3.fr)