

A289134


a(n) = 21*n^2  33*n + 13.


1



1, 31, 103, 217, 373, 571, 811, 1093, 1417, 1783, 2191, 2641, 3133, 3667, 4243, 4861, 5521, 6223, 6967, 7753, 8581, 9451, 10363, 11317, 12313, 13351, 14431, 15553, 16717, 17923, 19171, 20461, 21793, 23167, 24583, 26041, 27541, 29083, 30667, 32293
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OFFSET

1,2


COMMENTS

a(n) is the sum of all cells in a cellularautomatalike 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.
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.


LINKS

Colin Barker, Table of n, a(n) for n = 1..1000
D. R. Reynolds, Geometric graphic of t, a(t) for t=1...4
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 21*n^2  33*n + 13.  Omar E. Pol, Jun 25 2017
G.f.: x*(1 + 28*x + 13*x^2) / (1  x)^3.  Colin Barker, Jun 28 2017
a(n) = 3*a(n1)  3*a(n2) + a(n3) for n>3.  Colin Barker, Jul 29 2017


MATHEMATICA

hexgro[t_]:=7+4*6+5*6*(t2)+Sum[i*6*7, {i, t2}]


PROG

(PARI) Vec(x*(1 + 28*x + 13*x^2) / (1  x)^3 + O(x^60)) \\ Colin Barker, Jun 28 2017


CROSSREFS

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.
Sequence in context: A222550 A075936 A221902 * A103069 A206526 A141877
Adjacent sequences: A289131 A289132 A289133 * A289135 A289136 A289137


KEYWORD

nonn,easy


AUTHOR

Daniel Rockwitz Reynolds, Jun 25 2017


STATUS

approved



