A339488 - OEIS

%I #20 Mar 30 2021 14:59:08

%S 0,-9,0,135,576,1575,3456,6615,11520,18711,28800,42471,60480,83655,

%T 112896,149175,193536,247095,311040,386631,475200,578151,696960,

%U 833175,988416,1164375,1362816,1585575,1834560,2111751,2419200,2759031,3133440,3544695,3995136

%N 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.

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

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

%H Roger C. Alperin, <a href="https://doi.org/10.1080/07468342.1987.11973023">Heron's area formula</a>, The College Mathematics Journal 18(2), 137-138, 1987.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/heron.pdf">Two one-line proofs of Heron's Formula</a>, Jan. 2014; <a href="/A339488/a339488.pdf">Local copy</a>

%H Mark Levi, <a href="https://sinews.siam.org/Details-Page/a-simple-derivation-of-herons-formula-2">A simple derivation of Heron’s formula</a>, SIAM news, December 2020.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heron%27s_formula">Heron's formula</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>.

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

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

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

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

%Y Cf. A336900, A002415.

%K sign

%O 0,2

%A _Peter Luschny_, Dec 16 2020