A334277 Perimeters of almost-equilateral Heronian triangles.

12, 42, 156, 582, 2172, 8106, 30252, 112902, 421356, 1572522, 5868732, 21902406, 81740892, 305061162, 1138503756, 4248953862, 15857311692, 59180292906, 220863859932, 824275146822, 3076236727356, 11480671762602, 42846450323052, 159905129529606, 596774067795372, 2227191141651882

Offset: 1

Author

Wesley Ivan Hurt, May 20 2020

Keywords

• nonn
• easy

Examples

			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].
		

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).

Crossrefs

Cf. A001075.

Cf. A011945 (areas), this sequence (perimeters).

Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).

Programs

Mathematica

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

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