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%I #58 Apr 29 2020 16:11:12
%S 1,31,103,217,373,571,811,1093,1417,1783,2191,2641,3133,3667,4243,
%T 4861,5521,6223,6967,7753,8581,9451,10363,11317,12313,13351,14431,
%U 15553,16717,17923,19171,20461,21793,23167,24583,26041,27541,29083,30667,32293
%N a(n) = 21*n^2 - 33*n + 13.
%C a(n) is the sum of all cells in a cellular-automata-like hexagonal lattice growth from a single active seed, based upon whether each hexagonal unit is active plus how many active neighbors each cell is touching for all active cells in the lattice.
%C The initial hexagonal seed starts with a single 1 representing it is active and touching no active neighbors. In the next time step, all inactive hexagonal neighboring spaces in the surrounding hexagonal lattice which were touching the active seed via edges become active and all active cells are summed together based on whether they are active plus how many active neighbors they are touching via their edges. This continues for each time step with inactive neighbors touching active neighbors in the previous time step becoming active in the current step followed by the described summing.
%H Colin Barker, <a href="/A289134/b289134.txt">Table of n, a(n) for n = 1..1000</a>
%H D. R. Reynolds, <a href="https://www.buffaloschools.org/cms/lib/NY01913551/Centricity/ModuleInstance/104110/large/hexgro4wex.jpg">Geometric graphic of t, a(t) for t=1...4</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(1 + 28*x + 13*x^2) / (1 - x)^3. - _Colin Barker_, Jun 28 2017
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - _Colin Barker_, Jul 29 2017
%t hexgro[t_]:=7+4*6+5*6*(t-2)+Sum[i*6*7,{i,t-2}]; Table[hexgro[n],{n,40}]
%t LinearRecurrence[{3,-3,1},{1,31,103},40] (* _Harvey P. Dale_, Apr 23 2020 *)
%o (PARI) Vec(x*(1 + 28*x + 13*x^2) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Jun 28 2017
%Y Cf. A033574 (analog for square tiling, von Neumann neighborhood), A016922 (analog for square tiling, Moore neighborhood), A016923 (analog for cubic 3D tiling, Moore neighborhood), A064762.
%K nonn,easy
%O 1,2
%A _Daniel Rockwitz Reynolds_, Jun 25 2017
%E New Name from _Omar E. Pol_, Jun 25 2017
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