Story
I began studying the Collatz Conjecture
And noticed this pattern in the numbers that go to 1 in one odd step, like 5,10,20,21,40,42... and looke in up on OEIS and found this formula.
\$floor(sqrt(4*n + 1)) - 1\$
Which can plot these numbers in their natural order like so;
\$\frac{\left(8\cdot2^{\operatorname{floor}\left(\sqrt{4\operatorname{floor}\left(x\right)+1}\right)}-2^{\left(\operatorname{floor}\left(\sqrt{4\operatorname{floor}\left(x\right)+1}\right)-1-\operatorname{floor}\left(\frac{\left(4\operatorname{floor}\left(x\right)+1-\operatorname{floor}\left(\sqrt{4\operatorname{floor}\left(x\right)+1}\right)^2\right)}{2}\right)\right)}\right)}{3}\$
Then I looked at numbers going to 1 in two steps, like 3,6,12,13,24,26...
Where I found another pattern that I could not find a formula for on OEIS
long nth(int n){if(n>241)return -1;return (((1<<Y[n]+5)-(1<<1+Y[n]-((Z[n]&1)+Z[n]*3)))/3-(1<<Y[n]-2*X[n]-(2*(Z[n]&1)+Z[n]*3)))/3;}
With X[],Y[] and Z[]
being these lookup-tables
int[]X=new int[]{
0,
0,
0, 1,
0, 1,
0, 1, 2,
0, 1, 2, 0,
0, 1, 2, 3, 0, 0,
0, 1, 2, 3, 0, 1, 0,
0, 1, 2, 3, 4, 0, 1, 0, 1,
0, 1, 2, 3, 4, 0, 1, 2, 0, 1,
0, 1, 2, 3, 4, 5, 0, 1, 2, 0, 1, 2,
0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 0,
0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 0, 1, 2, 3, 0, 0,
0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 0,
0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 0, 1,
0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 0, 1, 2, 0, 1,
0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 0, 1, 2,
0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 0,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 0, 1, 2, 3, 1, 2
};
int[]Y=new int[]{
0,
1,
2, 2,
3, 3,
4, 4, 4,
5, 5, 5, 5,
6, 6, 6, 6, 6, 6,
7, 7, 7, 7, 7, 7, 7,
8, 8, 8, 8, 8, 8, 8, 8, 8,
9, 9, 9, 9, 9, 9, 9, 9, 9, 9,
10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11,
12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,
16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17,
18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18
};
int[]Z=new int[]{
0,
0,
0, 0,
0, 0,
0, 0, 0,
0, 0, 0, 1,
0, 0, 0, 0, 1, 2,
0, 0, 0, 0, 1, 1, 2,
0, 0, 0, 0, 0, 1, 1, 2, 2,
0, 0, 0, 0, 0, 1, 1, 1, 2, 2,
0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2,
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3,
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4,
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4,
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4,
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5
};
Challenge
The challenge is to write a "reasonably fast" function or expression that replaces and extends these lookup tables.
Think of the lookup tables as a 3D structure.
Pictured is the top 720 boxes of this structure.
Input
An integer which is the index of a cube in the structure.
You can assume the input will be in the range 0 to 719 inclusive.
Output
The x,y,z coordinates for the given index.
Assuming the input is between 0 and 719 the output ranges are
x, 0 to 13
y, 0 to 27
z, 0 to 8
It's fine to accept and return larger indexes correctly just not required.
Examples
i -> x y z
0 -> 0, 0, 0
12 -> 0, 5, 1
30 -> 4, 8, 0
65 -> 2, 11, 1
100 -> 0, 13, 2
270 -> 1, 19, 3
321 -> 1, 20, 6
719 -> 1, 27, 8
If you collapse the z-coordinate, then the structure is indexed top-down left right like shown below; Examples are marked in square brackets []
Y,Z 0,
0 | [0]
1 | 1
2 | 2 3
3 | 4 5
4 | 6 7 8 1,
5 | 9 10 11 |[12] 2,
6 | 13 14 15 16 | 17 | 18
7 | 19 20 21 22 | 23 24 | 25
8 | 26 27 28 29 [30] | 31 32 | 33 34
9 | 35 36 37 38 39 | 40 41 42 | 43 44
10 | 45 46 47 48 49 50 | 51 52 53 | 54 55 56 3,
11 | 57 58 59 60 61 62 | 63 64 [65] 66 | 67 68 69 | 70 4,
12 | 71 72 73 74 75 76 77 | 78 79 80 81 | 82 83 84 85 | 86 | 87
13 | 88 89 90 91 92 93 94 | 95 96 97 98 99 [100] 101 102 103 |104 105 |106
14 |107 108 109 110 111 112 113 114 |115 116 117 118 119 |120 121 122 123 124 |125 126 |127 128
15 |129 130 131 132 133 134 135 136 |137 138 139 140 141 142 |143 144 145 146 147 |148 149 150 |151 152
16 |153 154 155 156 157 158 159 160 161 |162 163 164 165 166 167 |168 169 170 171 172 173 |174 175 176 |177 178 179 5,
17 |180 181 182 183 184 185 186 187 188 |189 190 191 192 193 194 195 |196 197 198 199 200 201 |202 203 204 205 |206 207 208 |209 6,
18 |210 211 212 213 214 215 216 217 218 219 |220 221 222 223 224 225 226 |227 228 229 230 231 232 233 |234 235 236 237 |238 239 240 241 |242 |243
19 |244 245 246 247 248 249 250 251 252 253 |254 255 256 257 258 259 260 261 |262 263 264 265 266 267 268 |269[270]271 272 273 |274 275 276 277 |278 279 |280
20 |281 282 283 284 285 286 287 288 289 290 291 |292 293 294 295 296 297 298 299 |300 301 302 303 304 305 306 307 |308 309 310 311 312 |313 314 315 316 317 |318 319 |320[321]
X->| 0 1 2 3 4 5 6 7 8 9 10 | 0 1 2 3 4 5 6 7 | 0 1 2 3 4 5 6 7 | 0 1 2 3 4 | 0 1 2 3 4 | 0 1 | 0 1
Note that at even y-coordinates the structure expands in the x-direction,
and at 0 and 5 mod 6 in the z-direction.
Expect for the very top block.
Rules
This is code-golf, the shortest code in bytes wins.
Reasonably fast
As an additional requirement although not a competition of fastest code,
the code must still be shown to compute coordinates in a reasonable amount of time.
You may for example use try it online and run a loop through all coordinates under 720 without exceeding the time limit of a minute, printing is optional.
If you fail this rule, mark your answer with non competing
"storing information as you go" is forbidden.
For example executing f(100) should not depend on having computed f(99) previously.
Lookup tables are allowed but included in bytecount so aim to make them sparse if you choose to use them.
Example code
non-competing
coord coords(int index){
int a=0,b=0,c=0;
int x=0,y=0,z=0;
long n,k,one;
n = k = 3;
int t=0;
while(t<index){
int s=0;k++;n=k;
while(n>1 && s<4){ n/=n&-n;n=n*3+1; n/=n&-n;s++;}
if(s==2)t++;
}
n=k;
one=n&-n;k = one;while(k>1){k>>=1;c++;} n=3*n+one;
one=n&-n;k = one;while(k>1){k>>=1;b++;} n=3*n+one;
one=n&-n;k = one;while(k>1){k>>=1;a++;}
coord r;
r.x = (b-c-1)>>1;
r.y = a-5;
r.z = (a-b-2)/6 +(a-b-4)/6;
return r;
}
Try it online!