Table to illustrate proof of Kimberling’s “Even Second Column” conjecture

Red = congruent to 0 or 1 modulo 3.
* marks start of row in Kimberling’s array.

  nEven QZRep of nOdd QZRep of n
Block 0* 1[ 2][ 1]
Type A  2[ 0, 3][ 3]
Block 1  3[ 4][ 1, 3]
Type B* 4[ 2, 4][ 1, 4]
   5[ 0, 5][ 5]
Block 2  6[ 2, 5][ 1, 5]
Type A* 7[ 0, 3, 5][ 3, 5]
Block 3  8[ 6][ 1, 3, 5]
Type C* 9[ 2, 6][ 1, 6]
   10[ 0, 3, 6][ 3, 6]
Block 4* 11[ 4, 6][ 1, 3, 6]
Type B  12[ 2, 4, 6][ 1, 4, 6]
   13[ 0, 7][ 7]
Block 5  14[ 2, 7][ 1, 7]
Type A* 15[ 0, 3, 7][ 3, 7]
Block 6  16[ 4, 7][ 1, 3, 7]
Type B* 17[ 2, 4, 7][ 1, 4, 7]
   18[ 0, 5, 7][ 5, 7]
Block 7  19[ 2, 5, 7][ 1, 5, 7]
Type A* 20[ 0, 3, 5, 7][ 3, 5, 7]
Block 8  21[ 8][ 1, 3, 5, 7]
Type C* 22[ 2, 8][ 1, 8]
   23[ 0, 3, 8][ 3, 8]
Block 9  24[ 4, 8][ 1, 3, 8]
Type B* 25[ 2, 4, 8][ 1, 4, 8]
   26[ 0, 5, 8][ 5, 8]
Block 10* 27[ 2, 5, 8][ 1, 5, 8]
Type A  28[ 0, 3, 5, 8][ 3, 5, 8]
Block 11  29[ 6, 8][ 1, 3, 5, 8]
Type C* 30[ 2, 6, 8][ 1, 6, 8]
   31[ 0, 3, 6, 8][ 3, 6, 8]
Block 12  32[ 4, 6, 8][ 1, 3, 6, 8]
Type B* 33[ 2, 4, 6, 8][ 1, 4, 6, 8]
   34[ 0, 9][ 9]
Block 13* 35[ 2, 9][ 1, 9]
Type A  36[ 0, 3, 9][ 3, 9]
Block 14  37[ 4, 9][ 1, 3, 9]
Type B* 38[ 2, 4, 9][ 1, 4, 9]
   39[ 0, 5, 9][ 5, 9]
Block 15  40[ 2, 5, 9][ 1, 5, 9]
Type A* 41[ 0, 3, 5, 9][ 3, 5, 9]
Block 16  42[ 6, 9][ 1, 3, 5, 9]
Type C* 43[ 2, 6, 9][ 1, 6, 9]
   44[ 0, 3, 6, 9][ 3, 6, 9]
Block 17  45[ 4, 6, 9][ 1, 3, 6, 9]
Type B* 46[ 2, 4, 6, 9][ 1, 4, 6, 9]
   47[ 0, 7, 9][ 7, 9]
Block 18  48[ 2, 7, 9][ 1, 7, 9]
Type A* 49[ 0, 3, 7, 9][ 3, 7, 9]
Block 19  50[ 4, 7, 9][ 1, 3, 7, 9]
Type B* 51[ 2, 4, 7, 9][ 1, 4, 7, 9]
   52[ 0, 5, 7, 9][ 5, 7, 9]
Block 20* 53[ 2, 5, 7, 9][ 1, 5, 7, 9]
Type A  54[ 0, 3, 5, 7, 9][ 3, 5, 7, 9]
Block 21  55[ 10][ 1, 3, 5, 7, 9]
Type C  56[ 2, 10][ 1, 10]
 * 57[ 0, 3, 10][ 3, 10]
Block 22  58[ 4, 10][ 1, 3, 10]
Type B* 59[ 2, 4, 10][ 1, 4, 10]
   60[ 0, 5, 10][ 5, 10]
Block 23* 61[ 2, 5, 10][ 1, 5, 10]
Type A  62[ 0, 3, 5, 10][ 3, 5, 10]
Block 24  63[ 6, 10][ 1, 3, 5, 10]
Type C* 64[ 2, 6, 10][ 1, 6, 10]
   65[ 0, 3, 6, 10][ 3, 6, 10]
Block 25  66[ 4, 6, 10][ 1, 3, 6, 10]
Type B* 67[ 2, 4, 6, 10][ 1, 4, 6, 10]
   68[ 0, 7, 10][ 7, 10]
Block 26* 69[ 2, 7, 10][ 1, 7, 10]
Type A  70[ 0, 3, 7, 10][ 3, 7, 10]
Block 27  71[ 4, 7, 10][ 1, 3, 7, 10]
Type B* 72[ 2, 4, 7, 10][ 1, 4, 7, 10]
   73[ 0, 5, 7, 10][ 5, 7, 10]
Block 28  74[ 2, 5, 7, 10][ 1, 5, 7, 10]
Type A* 75[ 0, 3, 5, 7, 10][ 3, 5, 7, 10]
Block 29  76[ 8, 10][ 1, 3, 5, 7, 10]
Type C* 77[ 2, 8, 10][ 1, 8, 10]
   78[ 0, 3, 8, 10][ 3, 8, 10]
Block 30* 79[ 4, 8, 10][ 1, 3, 8, 10]
Type B  80[ 2, 4, 8, 10][ 1, 4, 8, 10]
   81[ 0, 5, 8, 10][ 5, 8, 10]
Block 31  82[ 2, 5, 8, 10][ 1, 5, 8, 10]
Type A* 83[ 0, 3, 5, 8, 10][ 3, 5, 8, 10]
Block 32  84[ 6, 8, 10][ 1, 3, 5, 8, 10]
Type C* 85[ 2, 6, 8, 10][ 1, 6, 8, 10]
   86[ 0, 3, 6, 8, 10][ 3, 6, 8, 10]
Block 33  87[ 4, 6, 8, 10][ 1, 3, 6, 8, 10]
Type B* 88[ 2, 4, 6, 8, 10][ 1, 4, 6, 8, 10]
   89[ 0, 11][ 11]
Block 34* 90[ 2, 11][ 1, 11]
Type A  91[ 0, 3, 11][ 3, 11]
Block 35  92[ 4, 11][ 1, 3, 11]
Type B* 93[ 2, 4, 11][ 1, 4, 11]
   94[ 0, 5, 11][ 5, 11]
Block 36* 95[ 2, 5, 11][ 1, 5, 11]
Type A  96[ 0, 3, 5, 11][ 3, 5, 11]
Block 37  97[ 6, 11][ 1, 3, 5, 11]
Type C  98[ 2, 6, 11][ 1, 6, 11]
 * 99[ 0, 3, 6, 11][ 3, 6, 11]
Block 38  100[ 4, 6, 11][ 1, 3, 6, 11]
Type B* 101[ 2, 4, 6, 11][ 1, 4, 6, 11]
   102[ 0, 7, 11][ 7, 11]
Block 39* 103[ 2, 7, 11][ 1, 7, 11]
Type A  104[ 0, 3, 7, 11][ 3, 7, 11]
Block 40  105[ 4, 7, 11][ 1, 3, 7, 11]
Type B* 106[ 2, 4, 7, 11][ 1, 4, 7, 11]
   107[ 0, 5, 7, 11][ 5, 7, 11]
Block 41  108[ 2, 5, 7, 11][ 1, 5, 7, 11]
Type A* 109[ 0, 3, 5, 7, 11][ 3, 5, 7, 11]
Block 42  110[ 8, 11][ 1, 3, 5, 7, 11]
Type C* 111[ 2, 8, 11][ 1, 8, 11]
   112[ 0, 3, 8, 11][ 3, 8, 11]
Block 43  113[ 4, 8, 11][ 1, 3, 8, 11]
Type B* 114[ 2, 4, 8, 11][ 1, 4, 8, 11]
   115[ 0, 5, 8, 11][ 5, 8, 11]
Block 44  116[ 2, 5, 8, 11][ 1, 5, 8, 11]
Type A* 117[ 0, 3, 5, 8, 11][ 3, 5, 8, 11]
Block 45  118[ 6, 8, 11][ 1, 3, 5, 8, 11]
Type C* 119[ 2, 6, 8, 11][ 1, 6, 8, 11]
   120[ 0, 3, 6, 8, 11][ 3, 6, 8, 11]
Block 46* 121[ 4, 6, 8, 11][ 1, 3, 6, 8, 11]
Type B  122[ 2, 4, 6, 8, 11][ 1, 4, 6, 8, 11]
   123[ 0, 9, 11][ 9, 11]
Block 47  124[ 2, 9, 11][ 1, 9, 11]
Type A* 125[ 0, 3, 9, 11][ 3, 9, 11]
Block 48  126[ 4, 9, 11][ 1, 3, 9, 11]
Type B* 127[ 2, 4, 9, 11][ 1, 4, 9, 11]
   128[ 0, 5, 9, 11][ 5, 9, 11]
Block 49  129[ 2, 5, 9, 11][ 1, 5, 9, 11]
Type A* 130[ 0, 3, 5, 9, 11][ 3, 5, 9, 11]
Block 50  131[ 6, 9, 11][ 1, 3, 5, 9, 11]
Type C* 132[ 2, 6, 9, 11][ 1, 6, 9, 11]
   133[ 0, 3, 6, 9, 11][ 3, 6, 9, 11]
Block 51  134[ 4, 6, 9, 11][ 1, 3, 6, 9, 11]
Type B* 135[ 2, 4, 6, 9, 11][ 1, 4, 6, 9, 11]
   136[ 0, 7, 9, 11][ 7, 9, 11]
Block 52* 137[ 2, 7, 9, 11][ 1, 7, 9, 11]
Type A  138[ 0, 3, 7, 9, 11][ 3, 7, 9, 11]
Block 53  139[ 4, 7, 9, 11][ 1, 3, 7, 9, 11]
Type B* 140[ 2, 4, 7, 9, 11][ 1, 4, 7, 9, 11]
   141[ 0, 5, 7, 9, 11][ 5, 7, 9, 11]
Block 54  142[ 2, 5, 7, 9, 11][ 1, 5, 7, 9, 11]
Type A* 143[ 0, 3, 5, 7, 9, 11][ 3, 5, 7, 9, 11]
Block 55  144[ 12][ 1, 3, 5, 7, 9, 11]
Type C* 145[ 2, 12][ 1, 12]
   146[ 0, 3, 12][ 3, 12]
Block 56  147[ 4, 12][ 1, 3, 12]
Type B* 148[ 2, 4, 12][ 1, 4, 12]
   149[ 0, 5, 12][ 5, 12]
Block 57  150[ 2, 5, 12][ 1, 5, 12]
Type A* 151[ 0, 3, 5, 12][ 3, 5, 12]
Block 58  152[ 6, 12][ 1, 3, 5, 12]
Type C* 153[ 2, 6, 12][ 1, 6, 12]
   154[ 0, 3, 6, 12][ 3, 6, 12]
Block 59  155[ 4, 6, 12][ 1, 3, 6, 12]
Type B* 156[ 2, 4, 6, 12][ 1, 4, 6, 12]
   157[ 0, 7, 12][ 7, 12]
Block 60* 158[ 2, 7, 12][ 1, 7, 12]
Type A  159[ 0, 3, 7, 12][ 3, 7, 12]
Block 61  160[ 4, 7, 12][ 1, 3, 7, 12]
Type B* 161[ 2, 4, 7, 12][ 1, 4, 7, 12]
   162[ 0, 5, 7, 12][ 5, 7, 12]
Block 62* 163[ 2, 5, 7, 12][ 1, 5, 7, 12]
Type A  164[ 0, 3, 5, 7, 12][ 3, 5, 7, 12]
Block 63  165[ 8, 12][ 1, 3, 5, 7, 12]
Type C  166[ 2, 8, 12][ 1, 8, 12]
 * 167[ 0, 3, 8, 12][ 3, 8, 12]
Block 64  168[ 4, 8, 12][ 1, 3, 8, 12]
Type B* 169[ 2, 4, 8, 12][ 1, 4, 8, 12]
   170[ 0, 5, 8, 12][ 5, 8, 12]
Block 65* 171[ 2, 5, 8, 12][ 1, 5, 8, 12]
Type A  172[ 0, 3, 5, 8, 12][ 3, 5, 8, 12]
Block 66  173[ 6, 8, 12][ 1, 3, 5, 8, 12]
Type C* 174[ 2, 6, 8, 12][ 1, 6, 8, 12]
   175[ 0, 3, 6, 8, 12][ 3, 6, 8, 12]
Block 67  176[ 4, 6, 8, 12][ 1, 3, 6, 8, 12]
Type B* 177[ 2, 4, 6, 8, 12][ 1, 4, 6, 8, 12]
   178[ 0, 9, 12][ 9, 12]
Block 68* 179[ 2, 9, 12][ 1, 9, 12]
Type A  180[ 0, 3, 9, 12][ 3, 9, 12]
Block 69  181[ 4, 9, 12][ 1, 3, 9, 12]
Type B* 182[ 2, 4, 9, 12][ 1, 4, 9, 12]
   183[ 0, 5, 9, 12][ 5, 9, 12]
Block 70  184[ 2, 5, 9, 12][ 1, 5, 9, 12]
Type A* 185[ 0, 3, 5, 9, 12][ 3, 5, 9, 12]
Block 71  186[ 6, 9, 12][ 1, 3, 5, 9, 12]
Type C* 187[ 2, 6, 9, 12][ 1, 6, 9, 12]
   188[ 0, 3, 6, 9, 12][ 3, 6, 9, 12]
Block 72* 189[ 4, 6, 9, 12][ 1, 3, 6, 9, 12]
Type B  190[ 2, 4, 6, 9, 12][ 1, 4, 6, 9, 12]
   191[ 0, 7, 9, 12][ 7, 9, 12]
Block 73  192[ 2, 7, 9, 12][ 1, 7, 9, 12]
Type A* 193[ 0, 3, 7, 9, 12][ 3, 7, 9, 12]
Block 74  194[ 4, 7, 9, 12][ 1, 3, 7, 9, 12]
Type B* 195[ 2, 4, 7, 9, 12][ 1, 4, 7, 9, 12]
   196[ 0, 5, 7, 9, 12][ 5, 7, 9, 12]
Block 75  197[ 2, 5, 7, 9, 12][ 1, 5, 7, 9, 12]
Type A* 198[ 0, 3, 5, 7, 9, 12][ 3, 5, 7, 9, 12]
Block 76  199[ 10, 12][ 1, 3, 5, 7, 9, 12]
Type C* 200[ 2, 10, 12][ 1, 10, 12]
   201[ 0, 3, 10, 12][ 3, 10, 12]
Block 77  202[ 4, 10, 12][ 1, 3, 10, 12]
Type B* 203[ 2, 4, 10, 12][ 1, 4, 10, 12]
   204[ 0, 5, 10, 12][ 5, 10, 12]
Block 78* 205[ 2, 5, 10, 12][ 1, 5, 10, 12]
Type A  206[ 0, 3, 5, 10, 12][ 3, 5, 10, 12]
Block 79  207[ 6, 10, 12][ 1, 3, 5, 10, 12]
Type C* 208[ 2, 6, 10, 12][ 1, 6, 10, 12]
   209[ 0, 3, 6, 10, 12][ 3, 6, 10, 12]
Block 80  210[ 4, 6, 10, 12][ 1, 3, 6, 10, 12]
Type B* 211[ 2, 4, 6, 10, 12][ 1, 4, 6, 10, 12]
   212[ 0, 7, 10, 12][ 7, 10, 12]
Block 81* 213[ 2, 7, 10, 12][ 1, 7, 10, 12]
Type A  214[ 0, 3, 7, 10, 12][ 3, 7, 10, 12]
Block 82  215[ 4, 7, 10, 12][ 1, 3, 7, 10, 12]
Type B* 216[ 2, 4, 7, 10, 12][ 1, 4, 7, 10, 12]
   217[ 0, 5, 7, 10, 12][ 5, 7, 10, 12]
Block 83  218[ 2, 5, 7, 10, 12][ 1, 5, 7, 10, 12]
Type A* 219[ 0, 3, 5, 7, 10, 12][ 3, 5, 7, 10, 12]
Block 84  220[ 8, 10, 12][ 1, 3, 5, 7, 10, 12]
Type C* 221[ 2, 8, 10, 12][ 1, 8, 10, 12]
   222[ 0, 3, 8, 10, 12][ 3, 8, 10, 12]
Block 85  223[ 4, 8, 10, 12][ 1, 3, 8, 10, 12]
Type B* 224[ 2, 4, 8, 10, 12][ 1, 4, 8, 10, 12]
   225[ 0, 5, 8, 10, 12][ 5, 8, 10, 12]
Block 86  226[ 2, 5, 8, 10, 12][ 1, 5, 8, 10, 12]
Type A* 227[ 0, 3, 5, 8, 10, 12][ 3, 5, 8, 10, 12]
Block 87  228[ 6, 8, 10, 12][ 1, 3, 5, 8, 10, 12]
Type C* 229[ 2, 6, 8, 10, 12][ 1, 6, 8, 10, 12]
   230[ 0, 3, 6, 8, 10, 12][ 3, 6, 8, 10, 12]
Block 88* 231[ 4, 6, 8, 10, 12][ 1, 3, 6, 8, 10, 12]
Type B  232[ 2, 4, 6, 8, 10, 12][ 1, 4, 6, 8, 10, 12]
   233[ 0, 13][ 13]
Block 89  234[ 2, 13][ 1, 13]
Type A* 235[ 0, 3, 13][ 3, 13]
Block 90  236[ 4, 13][ 1, 3, 13]
Type B* 237[ 2, 4, 13][ 1, 4, 13]
   238[ 0, 5, 13][ 5, 13]
Block 91* 239[ 2, 5, 13][ 1, 5, 13]
Type A  240[ 0, 3, 5, 13][ 3, 5, 13]
Block 92  241[ 6, 13][ 1, 3, 5, 13]
Type C* 242[ 2, 6, 13][ 1, 6, 13]
   243[ 0, 3, 6, 13][ 3, 6, 13]
Block 93  244[ 4, 6, 13][ 1, 3, 6, 13]
Type B* 245[ 2, 4, 6, 13][ 1, 4, 6, 13]
   246[ 0, 7, 13][ 7, 13]
Block 94* 247[ 2, 7, 13][ 1, 7, 13]
Type A  248[ 0, 3, 7, 13][ 3, 7, 13]
Block 95  249[ 4, 7, 13][ 1, 3, 7, 13]
Type B* 250[ 2, 4, 7, 13][ 1, 4, 7, 13]
   251[ 0, 5, 7, 13][ 5, 7, 13]
Block 96  252[ 2, 5, 7, 13][ 1, 5, 7, 13]
Type A* 253[ 0, 3, 5, 7, 13][ 3, 5, 7, 13]
Block 97  254[ 8, 13][ 1, 3, 5, 7, 13]
Type C* 255[ 2, 8, 13][ 1, 8, 13]
   256[ 0, 3, 8, 13][ 3, 8, 13]
Block 98  257[ 4, 8, 13][ 1, 3, 8, 13]
Type B* 258[ 2, 4, 8, 13][ 1, 4, 8, 13]
   259[ 0, 5, 8, 13][ 5, 8, 13]
Block 99  260[ 2, 5, 8, 13][ 1, 5, 8, 13]
Type A* 261[ 0, 3, 5, 8, 13][ 3, 5, 8, 13]