# Resistance Distance
The submatrix obtained from the Laplacian matrix $L$ by deleting its $i$-th row and the $i$-th column will be denoted by $L(i)$.
The submatrix obtained from the Laplacian matrix $L$ by deleting its $i$-th and $j$-th rows and the $i$-th and $j$-th columns will be denoted by $L(i, j)$, assuming that $i \neq j$.
The resistance distance between two vertices $i$ and $j$, denoted by $r_{i, j}$, has following theorem:
:::success
Let $G$ be a connected graph on $n$ vertices, $n \ge 3$, and $1 \le i \neq j \le n$. Then $$r_{i, j}=\dfrac{\det L(i, j)}{\det L(i
)}=\dfrac{\det L(i, j)}{t(G)},$$ where $t(G)$ is the number of spanning trees of $G$.
:::
---
The resistance between two vertices at distance $k$ (for $1 \le k \le n$) in an $n$-dimensional hypercube is $$R_{n, k}=\dfrac{2}{n}\sum_{0\le j\le k-1} \dfrac{1}{n-1 \choose j}\dfrac{1}{2^n}\sum_{j+1\le i\le n}{n \choose i}.$$
Specially, $$R_{n, 1}=\dfrac{2^n-1}{n\cdot 2^{n-1}}, \quad R_{n, n}=\dfrac{1}{n}\sum_{0 \le j \le n-1}\dfrac{1}{n-1 \choose j}.$$
### resistance Table
If we look at the denominator, it seems that we can still divide it into two groups based on parity. But I am still unable to find a simple explanation from the perspective of calculating resistance.
$\ \ \rightarrow$ distance
$\downarrow$ dimension
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|:---:|:-------------------:|:-------------------:|:----------------------:|:-------------------:|:-----------------------:|:---------------------:|:-----------------------:|:--------------------:|:-----------------:|
| 1 | $1$ | | | | | | | | |
| 2 | $\dfrac{3}{4}$ | $1$ | | | | | | | |
| 3 | $\dfrac{7}{12}$ | $\dfrac{3}{4}$ | $\dfrac{5}{6}$ | | | | | | |
| 4 | $\dfrac{15}{32}$ | $\dfrac{7}{12}$ | $\dfrac{61}{96}$ | $\dfrac{2}{3}$ | | | | | |
| 5 | $\dfrac{31}{80}$ | $\dfrac{15}{32}$ | $\dfrac{241}{480}$ | $\dfrac{25}{48}$ | $\dfrac{8}{15}$ | | | | |
| 6 | $\dfrac{21}{64}$ | $\dfrac{31}{80}$ | $\dfrac{131}{320}$ | $\dfrac{101}{240}$ | $\dfrac{137}{320}$ | $\dfrac{13}{30}$ | | | |
| 7 | $\dfrac{127}{448}$ | $\dfrac{21}{64}$ | $\dfrac{12}{35}$ | $\dfrac{7}{20}$ | $\dfrac{2381}{6720}$ | $\dfrac{343}{960}$ | $\dfrac{151}{420}$ | | |
| 8 | $\dfrac{255}{1024}$ | $\dfrac{127}{448}$ | $\dfrac{2105}{7168}$ | $\dfrac{167}{560}$ | $\dfrac{10781}{35840}$ | $\dfrac{2033}{6720}$ | $\dfrac{32663}{107520}$ | $\dfrac{32}{105}$ | |
| 9 | $\dfrac{511}{2304}$ | $\dfrac{255}{1024}$ | $\dfrac{16531}{64512}$ | $\dfrac{929}{3584}$ | $\dfrac{42061}{161280}$ | $\dfrac{9383}{35840}$ | $\dfrac{84677}{322560}$ | $\dfrac{2357}{8960}$ | $\dfrac{83}{315}$ |
---
---
---
# $N$ Table
$\ \ \rightarrow$ distance
$\downarrow$ dimension
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|:---:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:------:|
| 1 | 1 | | | | | | | | | | | | |
| 2 | 4 | 2 | | | | | | | | | | | |
| 3 | 24 | 8 | 6 | | | | | | | | | | |
| 4 | 96 | 24 | 96 | 12 | | | | | | | | | |
| 5 | 960 | 192 | 960 | 192 | 60 | | | | | | | | |
| 6 | 960 | 960 | 960 | 960 | 960 | 60 | | | | | | | |
| 7 | 13440 | 1920 | 13440 | 1920 | 13440 | 1920 | 420 | | | | | | |
| 8 | 107520 | 13440 | 107520 | 13440 | 107520 | 13440 | 107520 | 840 | | | | | |
| 9 | 645120 | 215040 | 645120 | 215040 | 645120 | 215040 | 645120 | 215040 | 2520 | | | | |
| 10 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 2520 | | | |
| 11 | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 27720 | | |
| 12 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 27720 | |
| 13 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 360360 |
給定dimension=n，不考慮距離為n的情況，則奇數距離的N皆相同、偶數距離的N皆相同。
例如當dimension=7，奇數距離的N皆為13440，偶數距離的N皆為1920。
給定dimension=n，距離為n的的情況，$N=\text{lcm}(1, 2, 3, \cdots, n)$
例如當dimension=8，距離為8時，$N=\text{lcm}(1, 2, 3, \cdots, 8)=840$
---
# gcd & N
在 n-cube 中，gcd sequence "大多"數為 $(1, n, 1, n, 1, n, ..., \text{lcm}(1, 2, \cdots, n))$
例如 8-cube 的 gcd sequence 為 (1, 8, 1, 8, 1, 8, 1, 128)
例外：6-cube, 9-cube, 10-cube, 12-cube
* 2-cube
| distance | 1 | 2 |
|:--------:|:---:|:---:|
| gcd | 1 | 2 |
| N | 4 | 2 |
* 3-cube
| distance | 1 | 2 | 3 |
|:--------:|:---:|:---:|:---:|
| gcd | 1 | 3 | 4 |
| N | 24 | 8 | 6 |
* 4-cube
| distance | 1 | 2 | 3 | 4 |
|:--------:|:---:|:---:|:---:|:---:|
| gcd | 1 | 4 | 1 | 8 |
| N | 96 | 24 | 96 | 12 |
* 5-cube
| distance | 1 | 2 | 3 | 4 | 5 |
|:--------:|:---:|:---:|:---:|:---:| --- |
| gcd | 1 | 5 | 1 | 5 | 16 |
| N | 960 | 192 | 960 | 192 | 60 |
* 6-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 |
|:--------:|:---:|:---:|:---:|:---:|:---:|:---:|
| gcd | 1 | 1 | 1 | 1 | 1 | 16 |
| N | 960 | 960 | 960 | 960 | 960 | 60 |
* 7-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|:--------:|:-----:|:----:|:-----:|:----:|:-----:|:----:|:---:|
| gcd | 1 | 7 | 1 | 7 | 1 | 7 | 32 |
| N | 13440 | 1920 | 13440 | 1920 | 13440 | 1920 | 420 |
* 8-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|:--------:|:------:|:-----:|:------:|:-----:|:------:|:-----:|:------:|:---:|
| gcd | 1 | 8 | 1 | 8 | 1 | 8 | 1 | 128 |
| N | 107520 | 13440 | 107520 | 13440 | 107520 | 13440 | 107520 | 840 |
* 9-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|:--------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:----:|
| gcd | 1 | 3 | 1 | 3 | 1 | 3 | 1 | 3 | 256 |
| N | 645120 | 215040 | 645120 | 215040 | 645120 | 215040 | 645120 | 215040 | 2520 |
* 10-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|:--------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:----:|
| gcd | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 256 |
| N | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 645120 | 2520 |
* 11-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|:--------:|:-------:|:------:|:-------:|:------:|:-------:|:------:|:-------:|:------:|:-------:|:------:|:-----:|
| gcd | 1 | 11 | 1 | 11 | 1 | 11 | 1 | 11 | 1 | 11 | 256 |
| N | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 7096320 | 645120 | 27720 |
* 12-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|:--------:|:--------:|:-------:|:--------:|:-------:|:--------:|:-------:|:--------:|:-------:|:--------:|:-------:|:--------:|:-----:|
| gcd | 1 | 4 | 1 | 4 | 1 | 4 | 1 | 4 | 1 | 4 | 1 | 1024 |
| N | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 7096320 | 28385280 | 27720 |
* 13-cube
| distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:---------:|:--------:|:------:|
| gcd | 1 | 13 | 1 | 13 | 1 | 13 | 1 | 13 | 1 | 13 | 1 | 13 | 2048 |
| N | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 738017280 | 56770560 | 360360 |
---
# Simplified graph
$V_i=\{\text{the n-tuples with }i\text{ 1's and } n-i \text{ 0's}\}$
* $V_0=\{(0, \cdots, 0)\}$
* $V_1=\{(1, 0, \cdots, 0), (0, 1, 0, \cdots, 0), \cdots, (0, \cdots, 0, 1)\}$
* ...
* $V_n=\{(1, \cdots, 1)\}$
The simplified graph for $n$-cube with distance $=n$：
and the number of edges between $V_i$ and $V_{i+1}$ is $$\binom{n}{i} (n-i)=n\binom{n-1}{i}$$
---
# Some examples of computation
## 2-cube (N=4)
firing script from simplified graph of distance 1 ： $(0, 3, 1, 2)$
$\rightarrow$ firing script for original graph of distance 1 ： $(0, 3, 1, 2)$
firing script for original graph of distance 2 ： $(0, 3, 1, 2)+(1, 0, 2, 3)=(1, 3, 3, 5) \sim (0, 2, 2, 4)$
Since $\text{gcd}(2, 2, 4)=2$, $N=\dfrac{4}{2}=2$.
---
## 3-cube (N=24)
firing script from simplified graph of distance 1 ： $(0, 14, 5, 9, 6, 8)$
$\rightarrow$ firing script for original graph of distance 1 ： $(0, 14, 5, 9, 5, 9, 6, 8)$
firing script for original graph of distance 2 ： $(0, 14, 5, 9, 5, 9, 6, 8)+(5, 0, 9, 14, 6, 5, 8, 9)=(5, 14, 14, 23, 11, 14, 14, 17) \sim (0, 9, 9, 18, 6, 9, 9, 12)$
Since $\text{gcd}(9, 9, 18, 6, 9, 9, 12)=3$, $N=\dfrac{24}{3}=8$.
firing script for original graph of distance 3 ： $(5, 14, 14, 23, 11, 14, 14, 17)+(6 ,5 ,5 ,0, 8, 9, 9, 14)=(11 ,19 ,19 ,23 ,19 ,23, 23 ,31) \sim (0, 8 ,8 ,12, 8 ,12, 12 ,20)$
Since $\text{gcd}(8 ,8 ,12, 8 ,12, 12 ,20)=4$, $N=\dfrac{24}{4}=6$.
---
## 4-cube (N=96)
$(0, 45, 17, 28, 20, 25, 21, 24) \rightarrow (0, 45, 17, 28, 17, 28, 20, 25, 17, 28, 20, 25, 20, 25, 21, 24)$
firing script for original graph of distance 2 ： $(0, 28, 28, 56, 20, 28, 28, 36, 20, 28, 28, 36, 24, 28, 28, 32)$
Since $\text{gcd}(28, 28, 56, 20, 28, 28, 36, 20, 28, 28, 36, 24, 28, 28, 32)=4$, $N=\dfrac{96}{4}=24$
firing script for original graph of distance 3 ： $(0, 25, 25, 36, 25, 36, 36, 61, 21, 28, 28, 33, 28, 33, 33, 40)$
Since $\text{gcd}(25, 25, 36, 25, 36, 36, 61, 21, 28, 28, 33, 28, 33, 33, 40)=1$, $N=\dfrac{96}{1}=96$.
firing script for original graph of distance 4 ： $(0, 24, 24, 32 ,24 ,32, 32 ,40, 24, 32, 32, 40, 32, 40, 40 ,64)$
Since $\text{gcd}(24, 24, 32 ,24 ,32, 32 ,40, 24, 32, 32, 40, 32, 40, 40 ,64)=8$, $N=\dfrac{96}{8}=12$
---
## 5-cube (N=960)
* dis=1
(0, 372, 147, 225, 147, 225, 170, 202, 147, 225, 170, 202, 170, 202, 177, 195, 147, 225, 170, 202, 170, 202, 177, 195, 170, 202, 177, 195, 177, 195, 180, 192)
gcd=1
* dis=2
(0, 225, 225, 450, 170, 225, 225, 280, 170, 225, 225, 280, 200, 225, 225, 250, 170, 225, 225, 280, 200, 225, 225, 250, 200, 225, 225, 250, 210, 225, 225, 240)
gcd=5
* dis=3
(0, 202, 202, 280, 202, 280, 280, 482, 177, 225, 225, 257, 225, 257, 257, 305, 177, 225, 225, 257, 225, 257, 257, 305, 210, 232, 232, 250, 232, 250, 250, 272)
gcd=1
* dis=4
(0, 195, 195, 250, 195, 250, 250, 305, 195, 250, 250, 305, 250, 305, 305, 500, 180, 225, 225, 250, 225, 250, 250, 275, 225, 250, 250, 275, 250, 275, 275, 320)
gcd=5
* dis=5
(0, 192, 192, 240, 192, 240, 240, 272, 192, 240, 240, 272, 240, 272, 272, 320, 192, 240, 240, 272, 240, 272, 272, 320, 240, 272, 272, 320, 272, 320, 320, 512)
gcd=16
---
## 6-cube (N=960)
* dis=1
(0, 315, 129, 186, 129, 186, 147, 168, 129, 186, 147, 168, 147, 168, 152, 163, 129, 186, 147, 168, 147, 168, 152, 163, 147, 168, 152, 163, 152, 163, 154, 161, 129, 186, 147, 168, 147, 168, 152, 163, 147, 168, 152, 163, 152, 163, 154, 161, 147, 168, 152, 163, 152, 163, 154, 161, 152, 163, 154, 161, 154, 161, 155, 160)
gcd=1
* dis=2
(0, 186, 186, 372, 147, 186, 186, 225, 147, 186, 186, 225, 170, 186, 186, 202, 147, 186, 186, 225, 170, 186, 186, 202, 170, 186, 186, 202, 177, 186, 186, 195, 147, 186, 186, 225, 170, 186, 186, 202, 170, 186, 186, 202, 177, 186, 186, 195, 170, 186, 186, 202, 177, 186, 186, 195, 177, 186, 186, 195, 180, 186, 186, 192)
gcd=1
* dis=3
(0, 168, 168, 225, 168, 225, 225, 393, 152, 186, 186, 207, 186, 207, 207, 241, 152, 186, 186, 207, 186, 207, 207, 241, 177, 191, 191, 202, 191, 202, 202, 216, 152, 186, 186, 207, 186, 207, 207, 241, 177, 191, 191, 202, 191, 202, 202, 216, 177, 191, 191, 202, 191, 202, 202, 216, 185, 193, 193, 200, 193, 200, 200, 208)
gcd=1
* dis=4
(0, 163, 163, 202, 163, 202, 202, 241, 163, 202, 202, 241, 202, 241, 241, 404, 154, 186, 186, 202, 186, 202, 202, 218, 186, 202, 202, 218, 202, 218, 218, 250, 154, 186, 186, 202, 186, 202, 202, 218, 186, 202, 202, 218, 202, 218, 218, 250, 180, 193, 193, 202, 193, 202, 202, 211, 193, 202, 202, 211, 202, 211, 211, 224)
gcd=1
* dis=5
(0, 161, 161, 195, 161, 195, 195, 216, 161, 195, 195, 216, 195, 216, 216, 250, 161, 195, 195, 216, 195, 216, 216, 250, 195, 216, 216, 250, 216, 250, 250, 411, 155, 186, 186, 200, 186, 200, 200, 211, 186, 200, 200, 211, 200, 211, 211, 225, 186, 200, 200, 211, 200, 211, 211, 225, 200, 211, 211, 225, 211, 225, 225, 256)
gcd=1
* dis=6
(0, 160, 160, 192, 160, 192, 192, 208, 160, 192, 192, 208, 192, 208, 208, 224, 160, 192, 192, 208, 192, 208, 208, 224, 192, 208, 208, 224, 208, 224, 224, 256, 160, 192, 192, 208, 192, 208, 208, 224, 192, 208, 208, 224, 208, 224, 224, 256, 192, 208, 208, 224, 208, 224, 224, 256, 208, 224, 224, 256, 224, 256, 256, 416)
gcd=16
---
## 7-cube (N=13440)
* dis=1
(0, 3810, 1605, 2205, 1605, 2205, 1806, 2004, 1605, 2205, 1806, 2004, 1806, 2004, 1857, 1953, 1605, 2205, 1806, 2004, 1806, 2004, 1857, 1953, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1605, 2205, 1806, 2004, 1806, 2004, 1857, 1953, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1857, 1953, 1876, 1934, 1876, 1934, 1885, 1925, 1605, 2205, 1806, 2004, 1806, 2004, 1857, 1953, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1857, 1953, 1876, 1934, 1876, 1934, 1885, 1925, 1806, 2004, 1857, 1953, 1857, 1953, 1876, 1934, 1857, 1953, 1876, 1934, 1876, 1934, 1885, 1925, 1857, 1953, 1876, 1934, 1876, 1934, 1885, 1925, 1876, 1934, 1885, 1925, 1885, 1925, 1890, 1920)
gcd=1
* dis=2
(0, 2205, 2205, 4410, 1806, 2205, 2205, 2604, 1806, 2205, 2205, 2604, 2058, 2205, 2205, 2352, 1806, 2205, 2205, 2604, 2058, 2205, 2205, 2352, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 1806, 2205, 2205, 2604, 2058, 2205, 2205, 2352, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 2128, 2205, 2205, 2282, 2156, 2205, 2205, 2254, 1806, 2205, 2205, 2604, 2058, 2205, 2205, 2352, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 2128, 2205, 2205, 2282, 2156, 2205, 2205, 2254, 2058, 2205, 2205, 2352, 2128, 2205, 2205, 2282, 2128, 2205, 2205, 2282, 2156, 2205, 2205, 2254, 2128, 2205, 2205, 2282, 2156, 2205, 2205, 2254, 2156, 2205, 2205, 2254, 2170, 2205, 2205, 2240)
gcd=7
* dis=3
(0, 2004, 2004, 2604, 2004, 2604, 2604, 4608, 1857, 2205, 2205, 2403, 2205, 2403, 2403, 2751, 1857, 2205, 2205, 2403, 2205, 2403, 2403, 2751, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 1857, 2205, 2205, 2403, 2205, 2403, 2403, 2751, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 2207, 2275, 2275, 2333, 2275, 2333, 2333, 2401, 1857, 2205, 2205, 2403, 2205, 2403, 2403, 2751, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 2207, 2275, 2275, 2333, 2275, 2333, 2333, 2401, 2128, 2256, 2256, 2352, 2256, 2352, 2352, 2480, 2207, 2275, 2275, 2333, 2275, 2333, 2333, 2401, 2207, 2275, 2275, 2333, 2275, 2333, 2333, 2401, 2240, 2284, 2284, 2324, 2284, 2324, 2324, 2368)
gcd=1
* dis=4
(0, 1953, 1953, 2352, 1953, 2352, 2352, 2751, 1953, 2352, 2352, 2751, 2352, 2751, 2751, 4704, 1876, 2205, 2205, 2352, 2205, 2352, 2352, 2499, 2205, 2352, 2352, 2499, 2352, 2499, 2499, 2828, 1876, 2205, 2205, 2352, 2205, 2352, 2352, 2499, 2205, 2352, 2352, 2499, 2352, 2499, 2499, 2828, 2156, 2275, 2275, 2352, 2275, 2352, 2352, 2429, 2275, 2352, 2352, 2429, 2352, 2429, 2429, 2548, 1876, 2205, 2205, 2352, 2205, 2352, 2352, 2499, 2205, 2352, 2352, 2499, 2352, 2499, 2499, 2828, 2156, 2275, 2275, 2352, 2275, 2352, 2352, 2429, 2275, 2352, 2352, 2429, 2352, 2429, 2429, 2548, 2156, 2275, 2275, 2352, 2275, 2352, 2352, 2429, 2275, 2352, 2352, 2429, 2352, 2429, 2429, 2548, 2240, 2303, 2303, 2352, 2303, 2352, 2352, 2401, 2303, 2352, 2352, 2401, 2352, 2401, 2401, 2464)
gcd=7
* dis=5
(0, 1934, 1934, 2282, 1934, 2282, 2282, 2480, 1934, 2282, 2282, 2480, 2282, 2480, 2480, 2828, 1934, 2282, 2282, 2480, 2282, 2480, 2480, 2828, 2282, 2480, 2480, 2828, 2480, 2828, 2828, 4762, 1885, 2205, 2205, 2333, 2205, 2333, 2333, 2429, 2205, 2333, 2333, 2429, 2333, 2429, 2429, 2557, 2205, 2333, 2333, 2429, 2333, 2429, 2429, 2557, 2333, 2429, 2429, 2557, 2429, 2557, 2557, 2877, 1885, 2205, 2205, 2333, 2205, 2333, 2333, 2429, 2205, 2333, 2333, 2429, 2333, 2429, 2429, 2557, 2205, 2333, 2333, 2429, 2333, 2429, 2429, 2557, 2333, 2429, 2429, 2557, 2429, 2557, 2557, 2877, 2170, 2284, 2284, 2352, 2284, 2352, 2352, 2410, 2284, 2352, 2352, 2410, 2352, 2410, 2410, 2478, 2284, 2352, 2352, 2410, 2352, 2410, 2410, 2478, 2352, 2410, 2410, 2478, 2410, 2478, 2478, 2592)
gcd=1
* dis=6
(0, 1925, 1925, 2254, 1925, 2254, 2254, 2401, 1925, 2254, 2254, 2401, 2254, 2401, 2401, 2548, 1925, 2254, 2254, 2401, 2254, 2401, 2401, 2548, 2254, 2401, 2401, 2548, 2401, 2548, 2548, 2877, 1925, 2254, 2254, 2401, 2254, 2401, 2401, 2548, 2254, 2401, 2401, 2548, 2401, 2548, 2548, 2877, 2254, 2401, 2401, 2548, 2401, 2548, 2548, 2877, 2401, 2548, 2548, 2877, 2548, 2877, 2877, 4802, 1890, 2205, 2205, 2324, 2205, 2324, 2324, 2401, 2205, 2324, 2324, 2401, 2324, 2401, 2401, 2478, 2205, 2324, 2324, 2401, 2324, 2401, 2401, 2478, 2324, 2401, 2401, 2478, 2401, 2478, 2478, 2597, 2205, 2324, 2324, 2401, 2324, 2401, 2401, 2478, 2324, 2401, 2401, 2478, 2401, 2478, 2478, 2597, 2324, 2401, 2401, 2478, 2401, 2478, 2478, 2597, 2401, 2478, 2478, 2597, 2478, 2597, 2597, 2912)
gcd=7
* dis=7
(0, 1920, 1920, 2240, 1920, 2240, 2240, 2368, 1920, 2240, 2240, 2368, 2240, 2368, 2368, 2464, 1920, 2240, 2240, 2368, 2240, 2368, 2368, 2464, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 1920, 2240, 2240, 2368, 2240, 2368, 2368, 2464, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 2368, 2464, 2464, 2592, 2464, 2592, 2592, 2912, 1920, 2240, 2240, 2368, 2240, 2368, 2368, 2464, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 2368, 2464, 2464, 2592, 2464, 2592, 2592, 2912, 2240, 2368, 2368, 2464, 2368, 2464, 2464, 2592, 2368, 2464, 2464, 2592, 2464, 2592, 2592, 2912, 2368, 2464, 2464, 2592, 2464, 2592, 2592, 2912, 2464, 2592, 2592, 2912, 2592, 2912, 2912, 4832)
gcd=32
---
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