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A335025 Largest side lengths of almost-equilateral Heronian triangles.

%I A335025 #20 Feb 16 2025 08:34:00
%S A335025 5,15,53,195,725,2703,10085,37635,140453,524175,1956245,7300803,
%T A335025 27246965,101687055,379501253,1416317955,5285770565,19726764303,
%U A335025 73621286645,274758382275,1025412242453,3826890587535,14282150107685,53301709843203,198924689265125,742397047217295,2770663499604053
%N A335025 Largest side lengths of almost-equilateral Heronian triangles.
%H A335025 Giovanni Resta, <a href="/A335025/b335025.txt">Table of n, a(n) for n = 1..1000</a>
%H A335025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%H A335025 Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%H A335025 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%H A335025 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F A335025 a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n + 1.
%F A335025 From _Alejandro J. Becerra Jr._, Feb 12 2021: (Start)
%F A335025 G.f.: x*(3*x^2 - 10*x + 5)/((1 - x)*(x^2 - 4*x + 1)).
%F A335025 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). (End)
%e A335025 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.
%e A335025 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.
%t A335025 Table[Expand[(2 + Sqrt[3])^n + (2 - Sqrt[3])^n + 1], {n, 40}]
%Y A335025 Cf. a(n) = A003500(n) + 1.
%Y A335025 Cf. A011945 (areas), A334277 (perimeters).
%Y A335025 Cf. A003500 (middle side lengths), A016064 (smallest side lengths), this sequence (largest side lengths).
%K A335025 nonn,easy
%O A335025 1,1
%A A335025 _Wesley Ivan Hurt_, May 20 2020