OFFSET
1,2
COMMENTS
On a hexagonal grid, cells are colored as follows: one cell and all those located along three straight lines passing through the center of the original cell and forming six 60° angles between each other are painted. In each of these corners, cells are painted over so that a V-shaped arrangement of cells repeats ad infinitum. The number of shaded cells in regular hexagons centered on the starting cell determines the sequence a(n).
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Members of the sequence a(1) - a(7).
Nicolay Avilov, Problem 2663. Snowflakes (in Russian).
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
Nicolay Avilov, Illustration a(13) and a(16)
FORMULA
a(n+4) = a(n) + 12*n + 18.
a(n) = 6*ceiling(n*(n - 1)/4) + 1.
a(n) = 6*A054925(n) + 1.
G.f.: (1 + 4*x - 4*x^2 + 4*x^3 + x^4)/((1 - x)^3*(1 + x^2)). - Stefano Spezia, Jun 11 2024
EXAMPLE
a(3) = 19 - 6*1 = 13;
a(4) = 37 - 6*3 = 19.
o . o . o
o . . o . o . . o .
o . o . o . o . o . o . o . o
o o . o o . . . o o . . . . . o o . . .
o o o o o o o o o o o o o o o o o o o o o o o o o
o o . o o . . . o o . . . . . o o . . .
o . o . o . o . o . o . o . o
o . . o . o . . o .
o . o . o
1 7 13 19 31
MATHEMATICA
Table[6*Ceiling[n*(n - 1)/4] + 1, {n, 100}] (* Paolo Xausa, Jul 01 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nicolay Avilov, Jun 10 2024
STATUS
approved