In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex polyiamonds with respect to their perimeter. In particular, we show that the number of

(A) baryiamonds of perimeter $n$ are asymptotically $\frac{{(\xi +1)}^2\sqrt{{\xi }^4+{\xi }^3-2\xi +1}}{2\sqrt{\pi n^3}}{\xi }^{-n-2},$where $\xi$ is a root of a certain explicit polynomial of degree 5.

(B) column-convex polyiamonds of perimeter $n$ are asymptotic to $\frac{(17997809\sqrt{17}+3^3\cdot 13\cdot 175463)\sqrt{95\sqrt{17}-119}}{2^7\cdot 43^2\cdot 89^2\sqrt{6\pi n^3}}{\left(\frac{3+\sqrt{17}}{2}\right)}^{n-1}.$

(C) convex polyiamonds of perimeter $n$ is asymptotic to $\frac{1280}{441\sqrt{3\pi n^3}}{3}^n.$

在本文中，我们考虑了居住在三角晶格上的动物的各种类别的聚钻石。通过对某些逐层分解和单元修剪/增长参数的仔细分析，我们得出了非空平移不变大麦芽（三角格子中的条形图），圆柱凸多菱形和凸出数量的生成函数的显式形式。 Polyiamonds的周边。特别是，我们表明

（A）周长 $ñ$ 渐近地 $\frac{{（\xi +\mathrm{1个}）}^2\sqrt{{\xi }^4+{\xi }^3-2\xi +\mathrm{1个}}}{2\sqrt{\pi {ñ}^3}}{\xi }^{-ñ-2}，$哪里 $\xi$ 是某个次数为5的显式多项式的根。

（B）周长的圆柱凸多边形 $ñ$ 渐近 $\frac{（17997809\sqrt{17}+3^3\cdot 13\cdot 175463）\sqrt{95\sqrt{17}-119}}{2^7\cdot 43^2\cdot 89^2\sqrt{6\pi {ñ}^3}}{\left(\frac{3+\sqrt{17}}{2}\right)}^{ñ-\mathrm{1个}}。$

（C）周边的凸多边形 $ñ$ 渐近于 $\frac{1280}{441\sqrt{3\pi {ñ}^3}}{3}^{ñ}。$