A335025 Largest side lengths of almost-equilateral Heronian triangles.

5, 15, 53, 195, 725, 2703, 10085, 37635, 140453, 524175, 1956245, 7300803, 27246965, 101687055, 379501253, 1416317955, 5285770565, 19726764303, 73621286645, 274758382275, 1025412242453, 3826890587535, 14282150107685, 53301709843203, 198924689265125, 742397047217295, 2770663499604053

Offset: 1

Author

Wesley Ivan Hurt, May 20 2020

Keywords

• nonn
• easy

Examples

			a(1) = 5; there is one Heronian triangle with perimeter 12 whose side lengths are consecutive integers, [3,4,5] and 5 is the largest side length.
a(2) = 15; there is one Heronian triangle with perimeter 42 whose side lengths are consecutive integers, [13,14,15] and 15 is the largest side length.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Heronian Triangle
• Wikipedia, Heronian triangle
• Wikipedia, Integer Triangle
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

Cf. a(n) = A003500(n) + 1.

Cf. A011945 (areas), A334277 (perimeters).

Cf. A003500 (middle side lengths), A016064 (smallest side lengths), this sequence (largest side lengths).

Programs

Mathematica

Table[Expand[(2 + Sqrt[3])^n + (2 - Sqrt[3])^n + 1], {n, 40}]

Formula

a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n + 1.

From Alejandro J. Becerra Jr., Feb 12 2021: (Start)

G.f.: x*(3*x^2 - 10*x + 5)/((1 - x)*(x^2 - 4*x + 1)).

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). (End)