An explicit formula for FM(1+1+n)

摘要

We show that the number of elements inFM(1+1+n), the modular lattice freely generated by two single elements and ann-element chain, is1$$frac{1}{{6sqrt 2 }}sumlimits_{k = 0}^{n + 1} {left {2left( {begin{array}{*{20}c} {2k} k end{array} } right) - left( {begin{array}{*{20}c} {2k} {k - 2} end{array} } right)} right} left( {lambda _1^{n - k + 2} - lambda _2^{n - k + 2} } right) - 2$$, where$$lambda _{1,2} = {{left( {4 pm 3sqrt 2 } right)} mathord{left/ {vphantom {{left( {4 pm 3sqrt 2 } right)} 2}} right. kern-nulldelimiterspace} 2}$$.