A270700 Triangular Star of David numbers (the figurate number of triangles framing a hexagram: a(0) = 12; thereafter a(n) = 36*n + 6).

12, 42, 78, 114, 150, 186, 222, 258, 294, 330, 366, 402, 438, 474, 510, 546, 582, 618, 654, 690, 726, 762, 798, 834, 870, 906, 942, 978, 1014, 1050, 1086, 1122, 1158, 1194, 1230, 1266, 1302, 1338, 1374, 1410, 1446, 1482, 1518, 1554, 1590, 1626, 1662, 1698, 1734

Offset: 0

Author

Peter M. Chema, Mar 21 2016

Keywords

• nonn
• easy

Comments

Also known as unitary triangular hexagram numbers, according to the author.

After a(0), the sum of inner and outer perimeters of triangle edges forming each hexagram is [36n - 6], always 12 less than the number of triangles framing the hexagram. Where a(0)=12, the perimeter is also 12.

Compare with A270545, the number of equilateral triangle units forming perimeters of equilateral triangle, which follows the same application.

Examples

			Illustration of initial terms are found in the three above links.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Peter M. Chema, Illustration of a(2)=78.
• Peter M. Chema, Illustration of initial terms [0 through 5].
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A045943, A045945, A045946, A270545, A271114.

Programs

Magma

[12] cat [36*n + 6: n in [1..50]]; // Vincenzo Librandi, Mar 28 2016

Mathematica

CoefficientList[Series[6 (1 + x) (2 + x)/(1 - x)^2, {x, 0, 40}], x] (* Michael De Vlieger, Mar 23 2016 *)
Join[{12},36*Range[50]+6] (* or *) LinearRecurrence[{2,-1},{12,42,78},50] (* Harvey P. Dale, Nov 03 2016 *)

PARI

a(n) = if (!n, 12, 36*n + 6); \\ Michel Marcus, Mar 22 2016

PARI

Vec(6*(1+x)*(2+x)/(1-x)^2 + O(x^50)) \\ Colin Barker, Mar 22 2016

Formula

a(0) = 12; thereafter, a(n) = 36*n + 6.

From Colin Barker, Mar 22 2016: (Start)

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

G.f.: 6*(1+x)*(2+x)/(1-x)^2. (End)

From Elmo R. Oliveira, Apr 04 2025: (Start)

E.g.f.: 6*(exp(x)*(6*x + 1) + 1).

a(n) = 6*A271114(n). (End)

Extensions

More terms from Vincenzo Librandi, Mar 28 2016