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A289156 Largest area of triangles with integer sides and area = n times perimeter.

%I A289156 #26 Sep 01 2025 16:56:54
%S A289156 60,1224,8436,34320,103020,254040,546084,1060896,1907100,3224040,
%T A289156 5185620,8004144,11934156,17276280,24381060,33652800,45553404,
%U A289156 60606216,79399860,102592080,130913580,165171864,206255076,255135840,312875100,380625960,459637524,551258736
%N A289156 Largest area of triangles with integer sides and area = n times perimeter.
%H A289156 Ray Chandler, <a href="/A289156/b289156.txt">Table of n, a(n) for n = 1..5000</a> (first 100 terms from Zhining Yang)
%H A289156 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A289156 From _Colin Barker_, Jun 28 2017: (Start)
%F A289156 G.f.: 12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6.
%F A289156 a(n) = 4*n*(2*n^2 + 1)*(4*n^2 + 1).
%F A289156 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6. (End)
%F A289156 a(n) = A120573(2*n). - _Ray Chandler_, Jul 27 2017
%F A289156 From _Elmo R. Oliveira_, Sep 01 2025: (Start)
%F A289156 E.g.f.: 4*exp(x)*x*(15 + 138*x + 206*x^2 + 80*x^3 + 8*x^4).
%F A289156 a(n) = 12*A005900(n)*A053755(n) = A053755(n)*A007900(n)/2. (End)
%e A289156 For n = 4, a(4) = 34320 means for the largest triangles (a,b,c) = (66,4225,4289), the area is 34320 which is 4 times the perimeter 8580.
%t A289156 Table[4 n (2 n^2 + 1) (4 n^2 + 1), {n, 27}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {60, 1224, 8436, 34320, 103020, 254040}, 27] (* or *) Rest@ CoefficientList[Series[12 x (5 + 72 x + 166 x^2 + 72 x^3 + 5 x^4)/(1 - x)^6, {x, 0, 27}], x] (* _Michael De Vlieger_, Jul 03 2017 *)
%o A289156 (PARI) Vec(12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6 + O(x^30)) \\ _Colin Barker_, Jun 28 2017
%Y A289156 Cf. A007237, A120573, A188158, A228383, A289155.
%Y A289156 Cf. A005900, A007900, A053755.
%K A289156 nonn,easy,changed
%O A289156 1,1
%A A289156 _Zhining Yang_, Jun 26 2017