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有形數

有形數是可以排成有一定規律形狀的。有形數是畢達哥拉斯學派的關注重點之一,他們認為數和形有不可分割的關係。有形數都是自然數,它們可以用小石子堆砌。有形數是將數形象化的方法。

一般地,任意一个自然数都可以表示为n个n边形数的和。(此即費馬多邊形數定理

前幾個平面上的有形數為:(不考慮trivial case,也就是n為n邊形數的情形)

6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, ... (OEIS數列A090466

種類

有形數可依照該數能排成的形狀分成:

多邊形數、多面體數、中心多邊形數中心多面體數星數、角錐數、角柱數、多胞體數.......等

例子

三角形數

能排成三角形有形數

三角形數.

17个三角形數是

13610152128364555667891105120136153……(OEIS數列A000217

梯形數

能排成等腰梯形有形數

2 7 15
** ***
****
****
*****
******

15個梯形數為

2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345......(OEIS數列A005449

梯形数公式:(顶层数+底层数)×层数÷2

中心五邊形數

排成從中心延伸出去的五邊形

中心五邊形數

15項的中心五邊形數為

1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526......(OEIS數列A005891).

四角錐數

能堆成四角錐有形數

四角錐數的計算方式

13四角錐數

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819...... (OEIS數列A000330).

六角星數

能排成六角星有形數

1 13 37
* *
****
***
****
*
*
**
*******
******
*****
******
*******
**
*

前13個六角星數

1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937......(OEIS數列A003154

參見

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