login
A335025
Largest side lengths of almost-equilateral Heronian triangles.
5
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
1,1
LINKS
Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Integer Triangle
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)
EXAMPLE
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.
MATHEMATICA
Table[Expand[(2 + Sqrt[3])^n + (2 - Sqrt[3])^n + 1], {n, 40}]
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).
Sequence in context: A099834 A034537 A318493 * A149581 A149582 A149583
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 20 2020
STATUS
approved