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).

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, 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

Offset: 1

Author

Alex Ratushnyak, Oct 05 2016

Keywords

• nonn

Comments

Similar to A214526, but three-dimensional, and the core is 2 X 2 X 2 rather than 1 X 1.

The spiral begins as follows:

Level z=-2:

95 94 93 92 91 90

96 77 76 75 74 89

97 78 67 66 73 88

98 79 68 65 72 87

99 80 69 70 71 86

100 81 82 83 84 85

z=-1:

116 115 114 113 112 111

117 52 51 50 49 110

118 53 62 61 60 109

119 54 63 64 59 108

120 55 56 57 58 107

101 102 103 104 105 106

z=0:

137 136 135 134 133 132

138 39 38 37 48 131

139 40 3 2 47 130

140 41 4 1 46 129

121 42 43 44 45 128

122 123 124 125 126 127

z=1:

144 145 146 147 148 149

143 34 35 36 25 150

142 33 6 7 26 151

141 32 5 8 27 152

160 31 30 29 28 153

159 158 157 156 155 154

z=2:

165 166 167 168 169 170

164 21 22 23 24 171

163 20 11 12 13 172

162 19 10 9 14 173

161 18 17 16 15 174

180 179 178 177 176 175

z=3:

186 187 188 189 190 191

185 204 205 206 207 192

184 203 214 215 208 193

183 202 213 216 209 194

182 201 212 211 210 195

181 200 199 198 197 196

Algorithm sketch:

1. At every x-y plane the direction is clockwise if z > 0 and counterclockwise if z <= 0.

2. After an N*N cube is complete and we start building an M*M cube, M=N+2:

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.

2b. Between the first and last z-edges the spiral progresses according to item 1.

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.

Crossrefs

Cf. A214526.

Formula

abs( a(n) - a(n-1) ) = 1.