A289156 - OEIS

%I #21 Jul 27 2017 12:56:52

%S 60,1224,8436,34320,103020,254040,546084,1060896,1907100,3224040,

%T 5185620,8004144,11934156,17276280,24381060,33652800,45553404,

%U 60606216,79399860,102592080,130913580,165171864,206255076,255135840

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

%H Ray Chandler, <a href="/A289156/b289156.txt">Table of n, a(n) for n = 1..5000</a> (first 100 terms from Zhining Yang)

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F From _Colin Barker_, Jun 28 2017: (Start)

%F G.f.: 12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4) / (1 - x)^6.

%F a(n) = 4*n*(2*n^2 + 1)*(4*n^2 + 1).

%F 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.

%F (End)

%F a(n) = A120573(2n). - _Ray Chandler_, Jul 27 2017

%e 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 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 (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 Cf. A007237, A120573, A188158, A228383, A289155.

%K nonn,easy

%O 1,1

%A _Zhining Yang_, Jun 26 2017