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

0,1

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.

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).

Formula

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

a(n) = 2*a(n-1)-a(n-2) for n>2. - Colin Barker, Mar 22 2016

G.f.: 6*(1+x)*(2+x) / (1-x)^2. - Colin Barker, Mar 22 2016

Example

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

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 *)

Prog

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

Crossrefs

Cf. A270545, A045945, A045943 and A045946.

Sequence in context: A241854 A085798 A335150 * A282693 A045945 A210206

Adjacent sequences: A270697 A270698 A270699 * A270701 A270702 A270703

Keyword

nonn,easy

Author

Peter M. Chema, Mar 21 2016

Extensions

More terms from Vincenzo Librandi, Mar 28 2016

Status

approved