A299965 Number of triangles in a Star of David of size n.

0, 20, 118, 348, 764, 1420, 2370, 3668, 5368, 7524, 10190, 13420, 17268, 21788, 27034, 33060, 39920, 47668, 56358, 66044, 76780, 88620, 101618, 115828, 131304, 148100, 166270, 185868, 206948, 229564, 253770, 279620, 307168, 336468, 367574, 400540, 435420, 472268

Offset

0,2

Comments

In a Star of David of size n, there are A135453(n) "size=1" triangles and 2*A228887(n) "size>1" triangles. See formula.

The number of matchstick units is A045946.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

John King, Star a=6, 84 matches, 118 triangles

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 9*n^3 + 12*n^2 - n.

a(n) = A135453(n) + 2 * A228887(n).

From Colin Barker, Apr 04 2019: (Start)

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

(End)

Example

For n=1, there are 12 (size=1) + 6 (size=4) + 2 (size=9) = 20 triangles.

Prog

(PARI) concat(0, Vec(2*x*(10 - x)*(1 + 2*x) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Apr 04 2019

Crossrefs

Cf. A045946, A135453, A228887.

For the total number of triangles in a different arrangement, see A002717 (for triangular matchstick), A045949 (for hexagonal matchstick).

Sequence in context: A206368 A258667 A357042 * A244289 A293880 A121040

Adjacent sequences: A299962 A299963 A299964 * A299966 A299967 A299968

Keyword

nonn,easy

Author

John King, Feb 22 2018

Status

approved