Math “数起来”；广义地说：“是的。”不是0,1,2,9,10,11,99而是00,01,10,11,02,12,20,21,22,03,99,_Math_Count_Language Agnostic_Numbers_Sequence

Math “数起来”；广义地说：“是的。”不是0,1,2,9,10,11,99而是00,01,10,11,02,12,20,21,22,03,99,

math language-agnostic

Math “数起来”；广义地说：“是的。”不是0,1,2,9,10,11,99而是00,01,10,11,02,12,20,21,22,03,99,,math,count,language-agnostic,numbers,sequence,Math,Count,Language Agnostic,Numbers,Sequence,当计数器由固定数量的数字（标题中的2）组成时，标准计数通过从最低有效位递增到最高有效位并在溢出时重置来工作我想以不同的方式计数：10进制中的4位数字将在2进制中计数，直到溢出回到0000为止，但相反，在忽略之前计数的数字时，将基数增加到3进制，因此只需继续0002、0012、0020、0021、0022、0112、0120、0121、0122、0200，020120210211，…（固定！）和所有其他至少有一个2英寸的数字。当2222切换到base-4时，所有的4位组合中至少有一个3位在后面。

当计数器由固定数量的数字（标题中的2）组成时，标准计数通过从最低有效位递增到最高有效位并在溢出时重置来工作

我想以不同的方式计数：10进制中的4位数字将在2进制中计数，直到溢出回到

为止，但相反，在忽略之前计数的数字时，将基数增加到3进制，因此只需继续
0002、0012、0020、0021、0022、0112、0120、0121、0122、0200，020120210211，…
（固定！）和所有其他至少有一个2英寸的数字。当
2222
切换到base-4时，所有的4位组合中至少有一个3位在后面。最后，从
0000
到
9999
的所有数字都按此顺序排列，但顺序不同
这样，在序列的最后10%之前，9不会出现在任何地方
理论上如何实现这样一个计数器（没有原始数字存在1检查）？我可以很容易地跳转到这个排序的第n个元素，或者向后计数吗？有没有一个真正的名字而不是“广泛计算”呢

1：一个“原始数字存在检查”将计至base-2，当切换到base-3时，将检查所有生成的数字，以确保其中至少有on-2。切换到基本4（即
2222
-
0003
步骤）时，所有数字必须至少包含一个
3
。因此，在
2222
之后，数字
0000
，
0001
，
0002
被省略，因为它们缺少3，因此之前已经被枚举过。以此类推，base-N表示数字N-1必须至少出现一次。
我制作了一个电子表格来查找模式。虽然我的电子表格显示得更高，显示的模式更清晰，但下面显示的是以3为基数的模式和以4为基数的前几个模式
在“table”（抱歉在SO中缺少表格格式）中，d3到d0是4位数字，str是相同数字的字符串表示，b是当前的数字基数，d是没有前导零的深度数字。pos是列表中的位置。（dec）是显示编号的十进制表示形式，以正常方式解释基数，由于存在重复项，因此证明其无效
请检查表格，看我是否正确解释了你的要求
一些模式正在出现，例如在给定基底和基底的每个深度的条目数之间似乎存在某种指数关系。我现在没有时间花在这上面，但在第二天左右我有机会的时候，我会进一步编辑这个答案，除非有人比我更擅长
至于这个名字，我不知道。但是，请注意，您允许的位数在很大程度上影响结果的顺序。另一方面，理论上没有必要停在9点，数学将继续上升到你喜欢的任何基数；如果您愿意，您可以使用A、B、C等，就像我们通常使用十六进制计数一样。此延续仅受允许的符号数限制

d3
d2
d1
d0
str
b
d
pos
（dec）

0
0
0
0
0000
1
0
0

0
0
0
1
0001
2
1
1

0
0
1
0
0010
2
2
2

0
0
1
1
0011
2
3
3

0
1
0
0
0100
2
3
4
4

0
1
0
1
0101
2
3
5

0
1
1
0
0110
2
3
6

0
1
1
1
0111
2
3
7

1
0
0
0
1000
2
4
8

1
0
0
1
1001
2
4
9

1
0
1
0
1010
2
4
10

1
0
1
1
1011
2
4
11
11

1
1
0
0
1100
2
4
12

1
1
0
1
1101
2
4
13
13

1
1
1
0
1110
2
4
14
14

1
1
1
1
1111
2
4
15

0
0
0
2
0002
3
1
16
2

0
0
1
2
for(maxdigit=0; maxdigit<10; ++maxdigit) { for(digit1 = 0; digit1 <= maxdigit; ++digit1) { for(digit2 = 0; digit2 <= maxdigit; ++digit2) { for(digit3 = 0; digit3 <= maxdigit; ++digit3) { for(digit4 = 0; digit4 <= maxdigit; ++digit4) { if( digit1 < maxdigit && digit2 < maxdigit && digit3 < maxdigit && digit4 < maxdigit) continue; print( digit1, digit2, digit3, digit4); }}}} } 00 | 01 | 02 | 03 | 04 ----- | | | 10 11 | 12 | 13 | 14 ---------- | | 20 21 22 | 23 | 24 --------------- | 30 31 32 33 | 34 -------------------- 40 41 42 43 44 for(shell=0;shell<=9;++shell) { // Do the L shaped part for(x=0;x<shell;++x) { // The the y-leg, which has z=k for(y=0;y<shell;++y) print(x,y,shell); // Do the z-leg, which has y=k for(z=0;z<=shell;++z) print(x,shell,z); } // Now the x=shell face for(y=0;y<=shell;++y) for(z=0;z<=shell;++z) print(shell,y,z); } // solve the dim-dimensional problem for // prefix the output with prefix function solve(int shell,int dim,String prefix) { if(dim==1) { // only one solution with the last digit being the shell print(prefix+shell); return; } // loop through the first digit for(int x=0;x<shell;++x) { String prefix2 = prefix + x; solve(shell,dim-1,prefix2); } // Now to the x=k hypercube, // need to do this in a separate recursion String prefix2 = prefix + shell; hypercube(shell,dim-1,prefix2); } // all solutions in dim dimensions with a given prefix function hypercube(int shell,int dim,String prefix) { if(dim==1) { for(int x=0;x<=shell;++x) println(prefix+x); } else { for(int x=0;x<=shell;++x) { String prefix2 = prefix + x; hypercube(shell,dim-1,prefix2); } } } // Now call, here we do the 4 digit version for(shell=0;shell<=9;++shell) { solve(shell,4,""); } +-----+------+------+------+------+ | | - | 2 | 3 | 4 | +-----+------+------+------+------+ | 0 | 0000 | | | | | 1 | | 0001 | | | | 2 | | 0010 | 0002 | | | 3 | | 0011 | | 0003 | | 4 | | 0100 | | | | 5 | | 0101 | 0012 | | | 6 | | 0110 | 0020 | | | 7 | | 0111 | 0021 | 0013 | | 8 | | 1000 | 0022 | | | 9 | | 1001 | | | | 10 | | 1010 | | | | 11 | | 1011 | 0102 | 0023 | | 12 | | 1100 | | 0030 | | 13 | | 1101 | | 0031 | | 14 | | 1110 | 0112 | 0032 | | 15 | | 1111 | 0120 | 0033 | | 16 | | | 0121 | | | 17 | | | 0122 | | | 18 | | | 0200 | | | 19 | | | 0201 | 0103 | | 20 | | | 0202 | | | 21 | | | 0210 | | | 22 | | | 0211 | | | 23 | | | 0212 | 0113 | | 24 | | | 0220 | | | 25 | | | 0221 | | | 26 | | | 0222 | | | 27 | | | | 0123 | | 28 | | | | 0130 | | 29 | | | 1002 | 0131 | | 30 | | | | 0132 | | 31 | | | | 0133 | | 32 | | | 1012 | | | 33 | | | 1020 | | | 34 | | | 1021 | | | 35 | | | 1022 | 0203 | | 36 | | | | | | 37 | | | | | | 38 | | | 1102 | | | 39 | | | | 0213 | | 40 | | | | | | 41 | | | 1112 | | | 42 | | | 1120 | | | 43 | | | 1121 | 0223 | | 44 | | | 1122 | 0230 | | 45 | | | 1200 | 0231 | | 46 | | | 1201 | 0232 | | 47 | | | 1202 | 0233 | | 48 | | | 1210 | 0300 | | 49 | | | 1211 | 0301 | | 50 | | | 1212 | 0302 | | 51 | | | 1220 | 0303 | | 52 | | | 1221 | 0310 | | 53 | | | 1222 | 0311 | | 54 | | | 2000 | 0312 | | 55 | | | 2001 | 0313 | | 56 | | | 2002 | 0320 | | 57 | | | 2010 | 0321 | | 58 | | | 2011 | 0322 | | 59 | | | 2012 | 0323 | | 60 | | | 2020 | 0330 | | 61 | | | 2021 | 0331 | | 62 | | | 2022 | 0332 | | 63 | | | 2100 | 0333 | | 64 | | | 2101 | | | 65 | | | 2102 | | | 66 | | | 2110 | | | 67 | | | 2111 | 1003 | | 68 | | | 2112 | | | 69 | | | 2120 | | | 70 | | | 2121 | | | 71 | | | 2122 | 1013 | | 72 | | | 2200 | | | 73 | | | 2201 | | | 74 | | | 2202 | | | 75 | | | 2210 | 1023 | | 76 | | | 2211 | 1030 | | 77 | | | 2212 | 1031 | | 78 | | | 2220 | 1032 | | 79 | | | 2221 | 1033 | | 80 | | | 2222 | | | 81 | | | | | | 82 | | | | | | 83 | | | | 1103 | | 84 | | | | | | 85 | | | | | | 86 | | | | | | 87 | | | | 1113 | | 88 | | | | | | 89 | | | | | | 90 | | | | | | 91 | | | | 1123 | | 92 | | | | 1130 | | 93 | | | | 1131 | | 94 | | | | 1132 | | 95 | | | | 1133 | | 96 | | | | | | 97 | | | | | | 98 | | | | | | 99 | | | | 1203 | | 100 | | | | | | 101 | | | | | | 102 | | | | | | 103 | | | | 1213 | | 104 | | | | | | 105 | | | | | | 106 | | | | | | 107 | | | | 1223 | | 108 | | | | 1230 | | 109 | | | | 1231 | | 110 | | | | 1232 | | 111 | | | | 1233 | | 112 | | | | 1300 | | 113 | | | | 1301 | | 114 | | | | 1302 | | 115 | | | | 1303 | | 116 | | | | 1310 | | 117 | | | | 1311 | | 118 | | | | 1312 | | 119 | | | | 1313 | | 120 | | | | 1320 | | 121 | | | | 1321 | | 122 | | | | 1322 | | 123 | | | | 1323 | | 124 | | | | 1330 | | 125 | | | | 1331 | | 126 | | | | 1332 | | 127 | | | | 1333 | | 128 | | | | | | 129 | | | | | | 130 | | | | | | 131 | | | | 2003 | | 132 | | | | | | 133 | | | | | | 134 | | | | | | 135 | | | | 2013 | | 136 | | | | | | 137 | | | | | | 138 | | | | | | 139 | | | | 2023 | | 140 | | | | 2030 | | 141 | | | | 2031 | | 142 | | | | 2032 | | 143 | | | | 2033 | | 144 | | | | | | 145 | | | | | | 146 | | | | | | 147 | | | | 2103 | | 148 | | | | | | 149 | | | | | | 150 | | | | | | 151 | | | | 2113 | | 152 | | | | | | 153 | | | | | | 154 | | | | | | 155 | | | | 2123 | | 156 | | | | 2130 | | 157 | | | | 2131 | | 158 | | | | 2132 | | 159 | | | | 2133 | | 160 | | | | | | 161 | | | | | | 162 | | | | | | 163 | | | | 2203 | | 164 | | | | | | 165 | | | | | | 166 | | | | | | 167 | | | | 2213 | | 168 | | | | | | 169 | | | | | | 170 | | | | | | 171 | | | | 2223 | | 172 | | | | 2230 | | 173 | | | | 2231 | | 174 | | | | 2232 | | 175 | | | | 2233 | | 176 | | | | 2300 | | 177 | | | | 2301 | | 178 | | | | 2302 | | 179 | | | | 2303 | | 180 | | | | 2310 | | 181 | | | | 2311 | | 182 | | | | 2312 | | 183 | | | | 2313 | | 184 | | | | 2320 | | 185 | | | | 2321 | | 186 | | | | 2322 | | 187 | | | | 2323 | | 188 | | | | 2330 | | 189 | | | | 2331 | | 190 | | | | 2332 | | 191 | | | | 2333 | | 192 | | | | 3000 | | 193 | | | | 3001 | | 194 | | | | 3002 | | 195 | | | | 3003 | | 196 | | | | 3010 | | 197 | | | | 3011 | | 198 | | | | 3012 | | 199 | | | | 3013 | | 200 | | | | 3020 | | 201 | | | | 3021 | | 202 | | | | 3022 | | 203 | | | | 3023 | | 204 | | | | 3030 | | 205 | | | | 3031 | | 206 | | | | 3032 | | 207 | | | | 3033 | | 208 | | | | 3100 | | 209 | | | | 3101 | | 210 | | | | 3102 | | 211 | | | | 3103 | | 212 | | | | 3110 | | 213 | | | | 3111 | | 214 | | | | 3112 | | 215 | | | | 3113 | | 216 | | | | 3120 | | 217 | | | | 3121 | | 218 | | | | 3122 | | 219 | | | | 3123 | | 220 | | | | 3130 | | 221 | | | | 3131 | | 222 | | | | 3132 | | 223 | | | | 3133 | | 224 | | | | 3200 | | 225 | | | | 3201 | | 226 | | | | 3202 | | 227 | | | | 3203 | | 228 | | | | 3210 | | 229 | | | | 3211 | | 230 | | | | 3212 | | 231 | | | | 3213 | | 232 | | | | 3220 | | 233 | | | | 3221 | | 234 | | | | 3222 | | 235 | | | | 3223 | | 236 | | | | 3230 | | 237 | | | | 3231 | | 238 | | | | 3232 | | 239 | | | | 3233 | | 240 | | | | 3300 | | 241 | | | | 3301 | | 242 | | | | 3302 | | 243 | | | | 3303 | | 244 | | | | 3310 | | 245 | | | | 3311 | | 246 | | | | 3312 | | 247 | | | | 3313 | | 248 | | | | 3320 | | 249 | | | | 3321 | | 250 | | | | 3322 | | 251 | | | | 3323 | | 252 | | | | 3330 | | 253 | | | | 3331 | | 254 | | | | 3332 | | 255 | | | | 3333 | +-----+------+------+------+------+ +-----+----+----+----+----+----+----+----+----+----+----+----+ | | - | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | +-----+----+----+----+----+----+----+----+----+----+----+----+ | 0 | 00 | | | | | | | | | | | | 1 | | 01 | | | | | | | | | | | 2 | | 10 | 02 | | | | | | | | | | 3 | | 11 | | 03 | | | | | | | | | 4 | | | | | 04 | | | | | | | | 5 | | | 12 | | | 05 | | | | | | | 6 | | | 20 | | | | 06 | | | | | | 7 | | | 21 | 13 | | | | 07 | | | | | 8 | | | 22 | | | | | | 08 | | | | 9 | | | | | 14 | | | | | 09 | | | 10 | | | | | | | | | | | 0A | | 11 | | | | 23 | | 15 | | | | | | | 12 | | | | 30 | | | | | | | | | 13 | | | | 31 | | | 16 | | | | | | 14 | | | | 32 | 24 | | | | | | | | 15 | | | | 33 | | | | 17 | | | | | 16 | | | | | | | | | | | | | 17 | | | | | | 25 | | | 18 | | | | 18 | | | | | | | | | | | | | 19 | | | | | 34 | | | | | 19 | | | 20 | | | | | 40 | | 26 | | | | | | 21 | | | | | 41 | | | | | | 1A | | 22 | | | | | 42 | | | | | | | | 23 | | | | | 43 | 35 | | 27 | | | | | 24 | | | | | 44 | | | | | | | | 25 | | | | | | | | | | | | | 26 | | | | | | | | | 28 | | | | 27 | | | | | | | 36 | | | | | | 28 | | | | | | | | | | | | | 29 | | | | | | 45 | | | | 29 | | | 30 | | | | | | 50 | | | | | | | 31 | | | | | | 51 | | 37 | | | | | 32 | | | | | | 52 | | | | | 2A | | 33 | | | | | | 53 | | | | | | | 34 | | | | | | 54 | 46 | | | | | | 35 | | | | | | 55 | | | 38 | | | | 36 | | | | | | | | | | | | | 37 | | | | | | | | | | | | | 38 | | | | | | | | | | | | | 39 | | | | | | | | 47 | | 39 | | | 40 | | | | | | | | | | | | | 41 | | | | | | | 56 | | | | | | 42 | | | | | | | 60 | | | | | | 43 | | | | | | | 61 | | | | 3A | | 44 | | | | | | | 62 | | 48 | | | | 45 | | | | | | | 63 | | | | | | 46 | | | | | | | 64 | | | | | | 47 | | | | | | | 65 | 57 | | | | | 48 | | | | | | | 66 | | | | | | 49 | | | | | | | | | | 49 | | | 50 | | | | | | | | | | | | | 51 | | | | | | | | | | | | | 52 | | | | | | | | | | | | | 53 | | | | | | | | | 58 | | | | 54 | | | | | | | | | | | 4A | | 55 | | | | | | | | 67 | | | | | 56 | | | | | | | | 70 | | | | | 57 | | | | | | | | 71 | | | | | 58 | | | | | | | | 72 | | | | | 59 | | | | | | | | 73 | | 59 | | | 60 | | | | | | | | 74 | | | | | 61 | | | | | | | | 75 | | | | | 62 | | | | | | | | 76 | 68 | | | | 63 | | | | | | | | 77 | | | | | 64 | | | | | | | | | | | | | 65 | | | | | | | | | | | 5A | | 66 | | | | | | | | | | | | | 67 | | | | | | | | | | | | | 68 | | | | | | | | | | | | | 69 | | | | | | | | | | 69 | | | 70 | | | | | | | | | | | | | 71 | | | | | | | | | 78 | | | | 72 | | | | | | | | | 80 | | | | 73 | | | | | | | | | 81 | | | | 74 | | | | | | | | | 82 | | | | 75 | | | | | | | | | 83 | | | | 76 | | | | | | | | | 84 | | 6A | | 77 | | | | | | | | | 85 | | | | 78 | | | | | | | | | 86 | | | | 79 | | | | | | | | | 87 | 79 | | | 80 | | | | | | | | | 88 | | | | 81 | | | | | | | | | | | | | 82 | | | | | | | | | | | | | 83 | | | | | | | | | | | | | 84 | | | | | | | | | | | | | 85 | | | | | | | | | | | | | 86 | | | | | | | | | | | | | 87 | | | | | | | | | | | 7A | | 88 | | | | | | | | | | | | | 89 | | | | | | | | | | 89 | | | 90 | | | | | | | | | | 90 | | | 91 | | | | | | | | | | 91 | | | 92 | | | | | | | | | | 92 | | | 93 | | | | | | | | | | 93 | | | 94 | | | | | | | | | | 94 | | | 95 | | | | | | | | | | 95 | | | 96 | | | | | | | | | | 96 | | | 97 | | | | | | | | | | 97 | | | 98 | | | | | | | | | | 98 | 8A | | 99 | | | | | | | | | | 99 | | | 100 | | | | | | | | | | | | | 101 | | | | | | | | | | | | | 102 | | | | | | | | | | | | | 103 | | | | | | | | | | | | | 104 | | | | | | | | | | | | | 105 | | | | | | | | | | | | | 106 | | | | | | | | | | | | | 107 | | | | | | | | | | | | | 108 | | | | | | | | | | | | | 109 | | | | | | | | | | | 9A | | 110 | | | | | | | | | | | A0 | | 111 | | | | | | | | | | | A1 | | 112 | | | | | | | | | | | A2 | | 113 | | | | | | | | | | | A3 | | 114 | | | | | | | | | | | A4 | | 115 | | | | | | | | | | | A5 | | 116 | | | | | | | | | | | A6 | | 117 | | | | | | | | | | | A7 | | 118 | | | | | | | | | | | A8 | | 119 | | | | | | | | | | | A9 | | 120 | | | | | | | | | | | AA | +-----+----+----+----+----+----+----+----+----+----+----+----+