Integers of form K*K + K*L + L*L arise a lot in connection with the triangular lattice, and so with the ring Z[tau], tau = exp(2πi/6) whose elements form a triangular lattice. A particularly nice way they arise is for hexagonal tessellations of the hexagonal torus T_H. T_H is defined by identifying opposite sides of a regular hexagon H. A hexagonal tessellation of T_H is just a tiling of T_H by congruent regular hexagons, as usual required to meet vertex to vertex and edge to edge. Then: ----- The number N is the number of tiles in some hexagonal tessellation of T_H if and only if For some K, L in Z: N = K*K + K*L + L*L. ----- Basic question: --------------- I made a table of the Loeschian numbers Lo(K,L) for K >= L. and then tried to *connect each one with the next largest one* with a line segment on paper. But it was hard to see any pattern in how they connect (below). So the question boils down to: Is there a concise formula for the "next largest number" in a set like Lo = {Lo(K,L) | K, L in Z} ??? I.e., f(0) = 1, f(1) = 3, f(3) = 4, ... in 0—1—3—4—7—9—12—13—16—19—21—25—27—28—31-36-39-43-48—49—.... ??? K| 0 1 2 3 4 5 6 7 8 9 10 L | ——————————————————————————————————————————————— 0 | 0 1 4 9 16 25 36 49 64 81 100 | 1 | 3 7 13 21 31 43 57 73 91 111 | 2 | 12 19 28 39 52 67 84 103 124 | 3 | 27 37 49 63 79 97 107 129 | 4 | 48 61 76 93 112 133 156 | 5 | 75 91 109 129 151 175 | 6 | 108 127 148 171 196 | 7 | 147 169 193 219 | 8 | 192 207 224 | 9 | 243 261 | 10 | 300 —Dan ————— PS The set Lo+ of numbers that are represented more than once by K*K + K*L + L*L (e.g. 49) has a particularly peculiar definition: OEIS A118886 says Lo+ consists of integers whose factorization into prime factors contains * at least two factors equal to 1 modulo 6, and * an even number of factors equal to -1 modulo 3.