A373584 a(n) is equal to the number of shaded cells in a regular hexagon with side n drawn on a hexagonal grid.

1, 7, 13, 19, 31, 49, 67, 85, 109, 139, 169, 199, 235, 277, 319, 361, 409, 463, 517, 571, 631, 697, 763, 829, 901, 979, 1057, 1135, 1219, 1309, 1399, 1489, 1585, 1687, 1789, 1891, 1999, 2113, 2227, 2341, 2461, 2587, 2713, 2839, 2971, 3109, 3247, 3385, 3529

Offset: 1

Author

Nicolay Avilov, Jun 10 2024

Keywords

• nonn
• easy
• changed

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).

Examples

			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
		

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).
• Nicolay Avilov, Illustration a(13) and a(16)
• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A003215, A011848, A054925.

Programs

Mathematica

Table[6*Ceiling[n*(n - 1)/4] + 1, {n, 100}] (* Paolo Xausa, Jul 01 2024 *)

Formula

a(n+4) = a(n) + 12*n + 18.

a(n) = 6*ceiling(n*(n - 1)/4) + 1.

a(n) = A003215(n) - 6*A011848(n+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

E.g.f.: (exp(x)*(5 + 6*x + 3*x^2) - 3*cos(x) + 3*sin(x))/2. - Stefano Spezia, Aug 31 2025