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

1,1

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}]

Crossrefs

Cf. A001075.

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

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

Sequence in context: A193068 A007586 A228391 * A122973 A074356 A172294

Adjacent sequences: A334274 A334275 A334276 * A334278 A334279 A334280

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 20 2020

Status

approved