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

0,3

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).

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

Mathematica

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

Prog

(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

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).

Sequence in context: A252255 A025415 A371420 * A125849 A003695 A022674

Adjacent sequences: A307250 A307251 A307252 * A307254 A307255 A307256

Keyword

nonn

Author

John King, Mar 31 2019

Status

approved