A339488 a(n) = H(n-1, n, n+1) where H(a, b, c) = (a + b + c)*(a + b - c)*(b + c - a)*(c + a - b) is Heron's polynomial.

0, -9, 0, 135, 576, 1575, 3456, 6615, 11520, 18711, 28800, 42471, 60480, 83655, 112896, 149175, 193536, 247095, 311040, 386631, 475200, 578151, 696960, 833175, 988416, 1164375, 1362816, 1585575, 1834560, 2111751, 2419200, 2759031, 3133440, 3544695, 3995136

Offset

0,2

Comments

The term 'Heron's polynomial' is not standard but inspired by Roger Alperin's proof of Heron's formula.

References

Reuben Hersh, Experiencing mathematics: What do we do, when we do mathematics?, p. 107, 2014.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Roger C. Alperin, Heron's area formula, The College Mathematics Journal 18(2), 137-138, 1987.

Shalosh B. Ekhad and Doron Zeilberger, Two one-line proofs of Heron's Formula, Jan. 2014; Local copy

Mark Levi, A simple derivation of Heron’s formula, SIAM news, December 2020.

Wikipedia, Heron's formula.

Wikipedia, Heronian triangle.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 3*n^4 - 12*n^2.

a(n) = [x^n] 9*x*(x + 1)*(x^2 - 6*x + 1)/(x - 1)^5.

a(2*n)/(24)^2 = binomial(n^2, 2)/6 = A002415(n) for n >= 0.

a(n) == 0 (mod 9). - Hugo Pfoertner, Nov 18 2025

Maple

seq(3*n^2*(n^2 - 4), n=0..34);

Mathematica

A339488[n_] := 3*n^2*(n^2 - 4); Array[A339488, 50, 0] (* Paolo Xausa, Jul 04 2026 *)
(* Alternative: *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, -9, 0, 135, 576}, 50] (* Paolo Xausa, Jul 04 2026 *)

Crossrefs

Cf. A336900, A002415.

Sequence in context: A222396 A222516 A057403 * A157309 A215484 A177145

Adjacent sequences: A339485 A339486 A339487 * A339489 A339490 A339491

Keyword

sign,easy,changed

Author

Peter Luschny, Dec 16 2020

Status

approved