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A277214 a(n) is the Manhattan distance between n and 1 in a 3-dimensional cubic spiral of positive integers with 1..8 at the center (illustration in the comments).

%I A277214 #17 Aug 17 2018 11:02:20
%S A277214 0,1,2,1,2,3,2,1,2,3,4,3,4,3,4,3,4,5,4,5,6,5,4,5,4,3,2,3,2,3,4,3,4,5,
%T A277214 4,3,2,3,4,3,2,3,2,1,2,1,2,3,4,3,4,5,4,3,4,3,2,3,2,3,2,3,2,1,2
%N A277214 a(n) is the Manhattan distance between n and 1 in a 3-dimensional cubic spiral of positive integers with 1..8 at the center (illustration in the comments).
%C A277214 Similar to A214526, but three-dimensional, and the core is 2 X 2 X 2 rather than 1 X 1.
%C A277214 The spiral begins as follows:
%C A277214 Level z=-2:
%C A277214    95  94  93  92  91  90
%C A277214    96  77  76  75  74  89
%C A277214    97  78  67  66  73  88
%C A277214    98  79  68  65  72  87
%C A277214    99  80  69  70  71  86
%C A277214   100  81  82  83  84  85
%C A277214 z=-1:
%C A277214   116 115 114 113 112 111
%C A277214   117  52  51  50  49 110
%C A277214   118  53  62  61  60 109
%C A277214   119  54  63  64  59 108
%C A277214   120  55  56  57  58 107
%C A277214   101 102 103 104 105 106
%C A277214 z=0:
%C A277214   137 136 135 134 133 132
%C A277214   138  39  38  37  48 131
%C A277214   139  40   3   2  47 130
%C A277214   140  41   4   1  46 129
%C A277214   121  42  43  44  45 128
%C A277214   122 123 124 125 126 127
%C A277214 z=1:
%C A277214   144 145 146 147 148 149
%C A277214   143  34  35  36  25 150
%C A277214   142  33   6   7  26 151
%C A277214   141  32   5   8  27 152
%C A277214   160  31  30  29  28 153
%C A277214   159 158 157 156 155 154
%C A277214 z=2:
%C A277214   165 166 167 168 169 170
%C A277214   164  21  22  23  24 171
%C A277214   163  20  11  12  13 172
%C A277214   162  19  10   9  14 173
%C A277214   161  18  17  16  15 174
%C A277214   180 179 178 177 176 175
%C A277214 z=3:
%C A277214   186 187 188 189 190 191
%C A277214   185 204 205 206 207 192
%C A277214   184 203 214 215 208 193
%C A277214   183 202 213 216 209 194
%C A277214   182 201 212 211 210 195
%C A277214   181 200 199 198 197 196
%C A277214 Algorithm sketch:
%C A277214 1. At every x-y plane the direction is clockwise if z > 0 and counterclockwise if z <= 0.
%C A277214 2. After an N*N cube is complete and we start building an M*M cube, M=N+2:
%C A277214 2a. The spiral at the first new edge of the M*M cube progresses from center to edges, in the same way as the A214526 spiral, e.g., z=-2 in the illustration.
%C A277214 2b. Between the first and last z-edges the spiral progresses according to item 1.
%C A277214 2c. The spiral at the last new edge of the M*M cube progresses from edges to center, e.g., z=3 in the illustration.
%F A277214 abs( a(n) - a(n-1) ) = 1.
%Y A277214 Cf. A214526.
%K A277214 nonn
%O A277214 1,3
%A A277214 _Alex Ratushnyak_, Oct 05 2016