A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

1, 7, 13, 25, 43, 61, 85, 115, 145, 181, 223, 265, 313, 367, 421, 481, 547, 613, 685, 763, 841, 925, 1015, 1105, 1201, 1303, 1405, 1513, 1627, 1741, 1861, 1987, 2113, 2245, 2383, 2521, 2665, 2815, 2965, 3121, 3283, 3445, 3613, 3787, 3961, 4141, 4327, 4513, 4705, 4903, 5101, 5305, 5515, 5725

Offset

0,2

Comments

In contrast, A003154 (the star numbers) counts the hexagonal lattice points within a hexagram centered at a lattice point and with the vertices of the central hexagon at the six lattice points a given number of steps outward from the central point.

Besides the first term, the first differences are given by six times A004396.

A005448 (the centered triangular numbers) counts just the lattice points within one of the two triangles that make up the hexagram.

Links

Table of n, a(n) for n=0..53.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(n) = 6*ceiling(n*(n+1)/3) + 1.

a(n) = 6*A007980(n-1) + 1 for n >= 1.

a(n+1) - a(n) = 6*A004396(n+1).

a(3n) = A081272(n).

Example

Illustration of initial terms:
.
.                                 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             25
.

Mathematica

Table[6*Ceiling[n*(n + 1)/3] + 1, {n, 0, 60}] (* Amiram Eldar, Jul 28 2023 *)

Prog

(PARI) a(n) = 6*ceil(n*(n+1)/3) + 1; \\ Michel Marcus, Jun 14 2024

Crossrefs

Cf. A003154, A004396, A005448, A007980, A081272.

Sequence in context: A200270 A031887 A294943 * A111721 A213663 A060455

Adjacent sequences: A363077 A363078 A363079 * A363081 A363082 A363083

Keyword

nonn,easy

Author

Aaron David Fairbanks, May 17 2023

Status

approved