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

60, 1224, 8436, 34320, 103020, 254040, 546084, 1060896, 1907100, 3224040, 5185620, 8004144, 11934156, 17276280, 24381060, 33652800, 45553404, 60606216, 79399860, 102592080, 130913580, 165171864, 206255076, 255135840, 312875100, 380625960, 459637524, 551258736

Offset: 1

Author

Zhining Yang, Jun 26 2017

Keywords

• nonn
• easy
• changed

Examples

			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.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..5000 (first 100 terms from Zhining Yang)
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A007237, A120573, A188158, A228383, A289155.

Cf. A005900, A007900, A053755.

Programs

Mathematica

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 *)

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

Formula

From Colin Barker, Jun 28 2017: (Start)

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

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

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)

a(n) = A120573(2*n). - Ray Chandler, Jul 27 2017

From Elmo R. Oliveira, Sep 01 2025: (Start)

E.g.f.: 4*exp(x)*x*(15 + 138*x + 206*x^2 + 80*x^3 + 8*x^4).

a(n) = 12*A005900(n)*A053755(n) = A053755(n)*A007900(n)/2. (End)