A307253 Number of triangles larger than size=1 in a matchstick-made hexagon with side length n.

0, 0, 14, 62, 166, 346, 624, 1020, 1556, 2252, 3130, 4210, 5514, 7062, 8876, 10976, 13384, 16120, 19206, 22662, 26510, 30770, 35464, 40612, 46236, 52356, 58994, 66170, 73906, 82222, 91140, 100680, 110864, 121712, 133246, 145486, 158454, 172170, 186656, 201932

Offset: 0

Author

John King, Mar 31 2019

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A033581 (number of size=1 triangles), A045949 (total number of triangles).

The hexagon matchstick sequences are as follows: number of matchsticks: A045945; for T1 triangles: A033581; for larger triangles: this sequence and for total triangles: A045949. There are analogs for triangles (see A045943) and stars (see A045946).

Programs

Mathematica

LinearRecurrence[{3,-2,-2,3,-1},{0, 0, 14, 62, 166},166] (* Metin Sariyar, Oct 27 2019 *)

PARI

concat([0,0], Vec(2*x^2*(7 + 10*x + 4*x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Apr 02 2019

Formula

a(n) = floor(n*(14*n^2+9*n+2)/4)-6*n^2.

G.f.: 2*x^2*(4*x^2+10*x+7)/((x+1)*(x-1)^4).

a(n) = A045949(n) - A033581(n).

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>4. - Colin Barker, Apr 02 2019