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%I #55 Oct 27 2019 12:03:31
%S 0,0,14,62,166,346,624,1020,1556,2252,3130,4210,5514,7062,8876,10976,
%T 13384,16120,19206,22662,26510,30770,35464,40612,46236,52356,58994,
%U 66170,73906,82222,91140,100680,110864,121712,133246,145486,158454,172170,186656,201932
%N Number of triangles larger than size=1 in a matchstick-made hexagon with side length n.
%H Colin Barker, <a href="/A307253/b307253.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F a(n) = floor(n*(14*n^2+9*n+2)/4)-6*n^2.
%F G.f.: 2*x^2*(4*x^2+10*x+7)/((x+1)*(x-1)^4).
%F a(n) = A045949(n) - A033581(n).
%F 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
%t LinearRecurrence[{3,-2,-2,3,-1},{0, 0, 14, 62, 166},166] (* _Metin Sariyar_, Oct 27 2019 *)
%o (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
%Y Cf. A033581 (number of size=1 triangles), A045949 (total number of triangles).
%Y 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).
%K nonn
%O 0,3
%A _John King_, Mar 31 2019
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