Triamond

A polyiamond (also polyamond or simply iamond) is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial 'di-' looks like a Greek prefix meaning 'two-' (though diamond actually derives from Greek ἀδάμας - also the basis for the word "adamant"). The name was suggested by recreational mathematics writer Thomas H. O'Beirne in New Scientist 1961 number 1, page 164.

The basic combinatorial question is, How many different polyiamonds exist with a given number of cells? Like polyominoes, polyiamonds may be either free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation. One-sided polyiamonds distinguish reflections.

The number of free n-iamonds for n = 1, 2, 3, … is:

The number of free polyiamonds with holes is given by  ; the number of free polyiamonds without holes is given by  ; the number of fixed polyiamonds is given by  ; the number of one-sided polyiamonds is given by  .

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