OFFSET
1,2
COMMENTS
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.
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.
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 72.
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
G.f.: x*(1 + 28*x + 13*x^2) / (1 - x)^3. - Colin Barker, Jun 28 2017
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - Colin Barker, Jul 29 2017
E.g.f.: exp(x)*(13 - 12*x + 21*x^2) - 13. - Stefano Spezia, Oct 19 2025
MATHEMATICA
hexgro[t_]:=7+4*6+5*6*(t-2)+Sum[i*6*7, {i, t-2}]; Table[hexgro[n], {n, 40}]
LinearRecurrence[{3, -3, 1}, {1, 31, 103}, 40] (* Harvey P. Dale, Apr 23 2020 *)
PROG
(PARI) Vec(x*(1 + 28*x + 13*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jun 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Rockwitz Reynolds, Jun 25 2017
EXTENSIONS
New Name from Omar E. Pol, Jun 25 2017
STATUS
approved