Distances in a bee's honeycomb

Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length 1.

One particular cell is occupied by the queen bee. For a positive real number L, let B(L) count the cells with distance L from the queen bee cell (all distances are measured from centre to centre); you may assume that the honeycomb is large enough to accommodate for any distance we wish to consider. For example, B(√3) = 6, B(√21) = 12 and B(111 111 111) = 54.

Find the number of L ≤ 5·10¹¹ such that B(L) = 450.

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter