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

1,1

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).

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)

Example

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.

Mathematica

Table[4 n (2 n^2 + 1) (4 n^2 + 1), {n, 27}] (* Michael De Vlieger, Jul 03 2017 *)
(* Alternative: *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {60, 1224, 8436, 34320, 103020, 254040}, 27] (* Michael De Vlieger, Jul 03 2017 *)
(* Alternative: *)
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 *)

Prog

(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

Crossrefs

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

Cf. A005900, A007900, A053755.

Sequence in context: A105252 A269138 A146347 * A288960 A269196 A379607

Adjacent sequences: A289153 A289154 A289155 * A289157 A289158 A289159

Keyword

nonn,easy,changed

Author

Zhining Yang, Jun 26 2017

Status

approved