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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..34.

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.

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.

MAPLE

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

CROSSREFS

Cf. A336900, A002415.

Sequence in context: A222396 A222516 A057403 * A157309 A215484 A177145

Adjacent sequences:  A339485 A339486 A339487 * A339489 A339490 A339491

KEYWORD

sign

AUTHOR

Peter Luschny, Dec 16 2020

STATUS

approved

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Last modified June 20 14:32 EDT 2021. Contains 345165 sequences. (Running on oeis4.)