求坐标的通项公式，初一题！

nyy

如图，把自然数在坐标格点上排成一个方形螺旋，1在(0,0), 2在(1,0), 3在(1,1), ……，那么2004在_______.



答案(1,-22).

现在请求出通项公式，由坐标求n，或者由 n 求坐标。

nyy

1,9,25,49等完全平方数都在一条直线上！

王守恩

  n, x, (1),a,b,(2), y, n,
01, 0,  1, 1, 1, 1, 0,01,  01=(0,0)
02, 1,  3, 2, 1, 2, 0,02,
03, 1,  4, 1, 2, 4, 1,03,  03=(1,1)
04, 0,  4, 0, 1, 5, 1,04,
05,-1,  4, 0, 1, 6, 1,05,
06,-1,  5, 1, 0, 6, 0,06,
07,-1,  6, 1, 0, 6,-1,07,
08, 0,  8, 2, 1, 7,-1,08,
09, 1,10, 2, 1, 8,-1,09,
10, 2,12, 2, 1, 9,-1,10,
11, 2,13, 1, 2,11, 0,11,
12, 2,14, 1, 2,13, 1,12,
13, 2,15, 1, 2,15, 2,13,
14, 1,15, 0, 1,16, 2,14,
15, 0,15, 0, 1,17, 2,15,
16,-1,15, 0, 1,18, 2,16,  16=(-1,2)
17,-2,15, 0, 1,19, 2,17,
18,-2,16, 1, 0,19, 1,18,
19,-2,17, 1, 0,19, 0,19,
20,-2,18, 1, 0,19,-1,20,
......

a累加的和 = (1) = n + x
b累加的和 = (2) = n + y

a=1, 21, 0011, 222111, 00001111, 2222211111, 000000111111, 22222221111111, 0000000011111111,  222222222111111111, 00000000001111111111, 2222222222211111111111, 000000000000111111111111, ...
b=1, 12, 1100, 111222, 11110000, 1111122222, 111111000000, 11111112222222, 1111111100000000,  111111111222222222, 11111111110000000000, 1111111111122222222222, 111111111111000000000000, ...

Jack315

2024 的坐标为 (21, -22) 。

王守恩

  n, x, a, b, y,
01, 0, 0, 0, 0,  01=(0,0)
02, 1, +,0, 0,
03, 1, 0,+, 1,  03=(1,1)
04, 0, -, 0, 1,
05,-1, -, 0, 1,
06,-1, 0, -, 0,
07,-1, 0, -, -1,
08, 0,+, 0, -1,
09, 1,+, 0, -1,
10, 2,+, 0, -1,
11, 2, 0, +, 0,
12, 2, 0, +, 1,
13, 2, 0, +, 2,
14, 1, -, 0, 2,
15, 0, -, 0, 2,
16,-1, -, 0, 2,  16=(-1,2)
17,-2, -, 0, 2,
18,-2, 0, -, 1,
19,-2, 0, -, 0,
20,-2, 0, -, -1,
......

a累加的和 = x
b累加的和 = y

a=0, +0, - - 00, +++000, - - - - 0000, +++++00000, - - - - - - 000000, +++++++0000000, - - - - - - - - 00000000, +++++++++000000000, - - - - - - - - - - 0000000000, +++++++++++00000000000, ...
b=0, 0+, 00 - -, 000+++, 0000 - - - -, 00000+++++, 000000 - - - - - -, 0000000+++++++, 00000000 - - - - - - - -, 000000000+++++++++, 0000000000 - - - - - - - - - -, 00000000000+++++++++++, ...

nyy

（$x>0$)
右下角 `n(x,-x+1)=4x^2-4x+2`
右上角           `n(x,x)=4x^2-2x+1`
左上角        `n(-x,x)=4x^2+1`
左下角     `n(-x,-x)=4x^2+2x+1`
右下角左     `n(x,-x)=4x^2+4x+1`

例1：x=1时，五个值分别是2, 3, 5, 7, 9
例2：x=2时，五个值分别是10, 13, 17, 21, 25

王守恩

拐点是这样一串数。1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50, 57, 65, 73, 82, 91, 101, 111, 122, 133, 145, 157, 170, 183, 197, 211, 226, 241, 257, 273, 290, 307, 325, 343, 362, 381, 401, ...

1. Table[Ceiling[(n^2 + 1)/4], {n, 99}]

复制代码

nyy

nyy 发表于 2024-12-6 09:19
（$x>0$)
右下角 `n(x,-x+1)=4x^2-4x+2`
右上角           `n(x,x)=4x^2-2x+1`

估计只能分段函数表示了

hujunhua

根据楼上的角值，可以写出一个分段函数\[
n(x,y)=\begin{cases}4x^2+1+y-3x,&\text{if}&-x<y<x\\
4y^2+1-x-y,&\text{if}&-y≤x≤y\\
4x^2+1-x-y,&\text{if}&x≤y≤-x\\
4y^2+1+x-3y,&\text{if}&y≤x≤-y
\end{cases}\]
然后简并为\[
n(x,y)=4\max(x^2,y^2)+1-x-y+δ(2x-2y),δ(x,y)=\begin{cases}-1,&\text{if}&-x<y<x\\
1,&\text{if}&y≤x≤-y\\
0,&\text{if}&\text{otherwise}
\end{cases}\]

nyy

hujunhua 发表于 2024-12-6 14:42
根据楼上的角值，可以写出一个分段函数\[
n(x,y)=\begin{cases}4x^2+1+y-3x,&\text{if}&-x ...

我觉得把图形看成一圈套圈这样，这样更有利于写出来。