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A334277 Perimeters of almost-equilateral Heronian triangles.

%I A334277 #38 Feb 16 2025 08:34:00
%S A334277 12,42,156,582,2172,8106,30252,112902,421356,1572522,5868732,21902406,
%T A334277 81740892,305061162,1138503756,4248953862,15857311692,59180292906,
%U A334277 220863859932,824275146822,3076236727356,11480671762602,42846450323052,159905129529606,596774067795372,2227191141651882
%N A334277 Perimeters of almost-equilateral Heronian triangles.
%H A334277 Giovanni Resta, <a href="/A334277/b334277.txt">Table of n, a(n) for n = 1..1000</a>
%H A334277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%H A334277 Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%H A334277 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%H A334277 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F A334277 a(n) = 3*A003500(n).
%F A334277 a(n) = 3 * ((2 + sqrt(3))^n + (2 - sqrt(3))^n).
%F A334277 From _Alejandro J. Becerra Jr._, Jan 29 2021: (Start)
%F A334277 G.f.: -6*x*(x - 2)/(x^2 - 4*x + 1).
%F A334277 a(n) = 4*a(n-1) - a(n-2). (End)
%F A334277 a(n) = 6 * A001075(n). - _Joerg Arndt_, Jan 29 2021
%F A334277 E.g.f.: 6*(exp(2*x)*cosh(sqrt(3)*x) - 1). - _Stefano Spezia_, Jan 29 2021
%e A334277 a(1) = 12; there is one Heronian triangle with perimeter 12 whose side lengths are consecutive integers, [3,4,5].
%e A334277 a(2) = 42; there is one Heronian triangle with perimeter 42 whose side lengths are consecutive integers, [13,14,15].
%t A334277 Table[Expand[3 ((2 + Sqrt[3])^n + (2 - Sqrt[3])^n)], {n, 40}]
%Y A334277 Cf. A001075.
%Y A334277 Cf. A011945 (areas), this sequence (perimeters).
%Y A334277 Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).
%K A334277 nonn,easy
%O A334277 1,1
%A A334277 _Wesley Ivan Hurt_, May 20 2020