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

%I A270700 #59 Apr 04 2025 18:14:20
%S A270700 12,42,78,114,150,186,222,258,294,330,366,402,438,474,510,546,582,618,
%T A270700 654,690,726,762,798,834,870,906,942,978,1014,1050,1086,1122,1158,
%U A270700 1194,1230,1266,1302,1338,1374,1410,1446,1482,1518,1554,1590,1626,1662,1698,1734
%N A270700 Triangular Star of David numbers (the figurate number of triangles framing a hexagram: a(0) = 12; thereafter a(n) = 36*n + 6).
%C A270700 Also known as unitary triangular hexagram numbers, according to the author.
%C A270700 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.
%C A270700 Compare with A270545, the number of equilateral triangle units forming perimeters of equilateral triangle, which follows the same application.
%H A270700 Colin Barker, <a href="/A270700/b270700.txt">Table of n, a(n) for n = 0..1000</a>
%H A270700 Peter M. Chema, <a href="/A270700/a270700.pdf">Illustration of a(2)=78</a>.
%H A270700 Peter M. Chema, <a href="/A270700/a270700_2.png">Illustration of initial terms [0 through 5]</a>.
%H A270700 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A270700 a(0) = 12; thereafter, a(n) = 36*n + 6.
%F A270700 From _Colin Barker_, Mar 22 2016: (Start)
%F A270700 a(n) = 2*a(n-1) - a(n-2) for n > 2.
%F A270700 G.f.: 6*(1+x)*(2+x)/(1-x)^2. (End)
%F A270700 From _Elmo R. Oliveira_, Apr 04 2025: (Start)
%F A270700 E.g.f.: 6*(exp(x)*(6*x + 1) + 1).
%F A270700 a(n) = 6*A271114(n). (End)
%e A270700 Illustration of initial terms are found in the three above links.
%t A270700 CoefficientList[Series[6 (1 + x) (2 + x)/(1 - x)^2, {x, 0, 40}], x] (* _Michael De Vlieger_, Mar 23 2016 *)
%t A270700 Join[{12},36*Range[50]+6] (* or *) LinearRecurrence[{2,-1},{12,42,78},50] (* _Harvey P. Dale_, Nov 03 2016 *)
%o A270700 (PARI) a(n) = if (!n, 12, 36*n + 6); \\ _Michel Marcus_, Mar 22 2016
%o A270700 (PARI) Vec(6*(1+x)*(2+x)/(1-x)^2 + O(x^50)) \\ _Colin Barker_, Mar 22 2016
%o A270700 (Magma) [12] cat [36*n + 6: n in [1..50]]; // _Vincenzo Librandi_, Mar 28 2016
%Y A270700 Cf. A045943, A045945, A045946, A270545, A271114.
%K A270700 nonn,easy
%O A270700 0,1
%A A270700 _Peter M. Chema_, Mar 21 2016
%E A270700 More terms from _Vincenzo Librandi_, Mar 28 2016