A045945 Hexagonal matchstick numbers: a(n) = 3*n*(3*n+1).

0, 12, 42, 90, 156, 240, 342, 462, 600, 756, 930, 1122, 1332, 1560, 1806, 2070, 2352, 2652, 2970, 3306, 3660, 4032, 4422, 4830, 5256, 5700, 6162, 6642, 7140, 7656, 8190, 8742, 9312, 9900, 10506, 11130, 11772, 12432, 13110, 13806, 14520, 15252, 16002, 16770, 17556

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

This may also be construed as the number of line segments illustrating the isometric projection of a cube of side length n. Moreover, a(n) equals the number of rods making a cube of side length n+1 minus the number of rods making a cube of side length n. See the illustration in the links and formula below.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Peter M. Chema, Illustration of initial terms as the first difference of number of rods required to make a 3-D cube.
• Craig Knecht, Number of positions a frame shifted H1 hexagon can occupy in a hexagon of order n.
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000290, A005449, A033580, A059986.

The hexagon matchstick sequences are: Number of matchsticks: this sequence; size=1 triangles: A033581; larger triangles: A307253; total triangles: A045949. Analog for triangles: A045943; analog for stars: A045946. - John King, Apr 05 2019

Programs

Maple

a:= n-> 3*n*(3*n+1): seq(a(n), n=0..42); # Zerinvary Lajos, May 03 2007

Mathematica

f[n_]:=3*n*(3*n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)

PARI

a(n) = 3*n*(3*n+1) \\ Charles R Greathouse IV, Feb 27 2017

Python

def a(n): return 3*n*(3*n+1) # Indranil Ghosh, Mar 26 2017

Formula

a(n) = a(n-1) + 6*(3*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 18 2010

G.f.: 6*x*(2+x)/(1-x)^3. - Colin Barker, Feb 12 2012

a(n) = 6*A005449(n). - R. J. Mathar, Feb 13 2016

a(n) = A059986(n) - A059986(n-1). - Peter M. Chema, Feb 26 2017

a(n) = 6*(A000217(n) + A000290(n)). - Peter M. Chema, Mar 26 2017

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=1} 1/a(n) = 1 - Pi/(6*sqrt(3)) - log(3)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = -1 + Pi/(3*sqrt(3)) + 2*log(2)/3. (End)

From Elmo R. Oliveira, Dec 12 2024: (Start)

E.g.f.: 3*exp(x)*x*(4 + 3*x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)