OFFSET
1,1
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 (4,-1).
FORMULA
a(n) = 3*A003500(n).
a(n) = 3 * ((2 + sqrt(3))^n + (2 - sqrt(3))^n).
From Alejandro J. Becerra Jr., Jan 29 2021: (Start)
G.f.: -6*x*(x - 2)/(x^2 - 4*x + 1).
a(n) = 4*a(n-1) - a(n-2). (End)
a(n) = 6 * A001075(n). - Joerg Arndt, Jan 29 2021
E.g.f.: 6*(exp(2*x)*cosh(sqrt(3)*x) - 1). - Stefano Spezia, Jan 29 2021
EXAMPLE
a(1) = 12; there is one Heronian triangle with perimeter 12 whose side lengths are consecutive integers, [3,4,5].
a(2) = 42; there is one Heronian triangle with perimeter 42 whose side lengths are consecutive integers, [13,14,15].
MATHEMATICA
Table[Expand[3 ((2 + Sqrt[3])^n + (2 - Sqrt[3])^n)], {n, 40}]
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 20 2020
STATUS
approved