Mnogokotniško število: Razlika med redakcijama
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Vrstica 194: | Vrstica 194: | ||
| align="center" bgcolor="#A0E0A0" | '''Ime''' |
| align="center" bgcolor="#A0E0A0" | '''Ime''' |
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| align="center" bgcolor="#A0E0A0" | '''Enačba''' |
| align="center" bgcolor="#A0E0A0" | '''Enačba''' |
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| align="center" bgcolor="#A0E0A0" | ''' |
| align="center" bgcolor="#A0E0A0" | '''''n''=1''' |
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| align="center" bgcolor="#A0E0A0" | '''2''' |
| align="center" bgcolor="#A0E0A0" | '''2''' |
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| align="center" bgcolor="#A0E0A0" | '''3''' |
| align="center" bgcolor="#A0E0A0" | '''3''' |
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Vrstica 209: | Vrstica 209: | ||
|----- |
|----- |
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| align="center" | [[trikotniško število|Trikotniško]] |
| align="center" | [[trikotniško število|Trikotniško]] |
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| align="center" | ½ |
| align="center" | ½''n''(1''n'' + 1) |
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| align="right" | 1 || align="right" | 3 |
| align="right" | 1 || align="right" | 3 |
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| align="right" | 6 |
| align="right" | 6 |
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Vrstica 220: | Vrstica 220: | ||
|----- |
|----- |
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| align="center" | [[kvadratno število|Kvadratno]] |
| align="center" | [[kvadratno število|Kvadratno]] |
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| align="center" | ½ |
| align="center" | ½''n''(2''n'' - 0) |
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| align="right" | 1 || align="right" | 4 |
| align="right" | 1 || align="right" | 4 |
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| align="right" | 9 |
| align="right" | 9 |
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Vrstica 231: | Vrstica 231: | ||
|----- |
|----- |
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| align="center" | [[petkotniško število|Petkotniško]] |
| align="center" | [[petkotniško število|Petkotniško]] |
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| align="center" | ½ |
| align="center" | ½''n''(3''n'' - 1) |
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| align="right" | 1 || align="right" | 5 |
| align="right" | 1 || align="right" | 5 |
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| align="right" | 12 |
| align="right" | 12 |
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Vrstica 242: | Vrstica 242: | ||
|----- |
|----- |
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| align="center" | [[šestkotniško število|Šestkotniško]] |
| align="center" | [[šestkotniško število|Šestkotniško]] |
||
| align="center" | ½ |
| align="center" | ½''n''(4''n'' - 2) |
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| align="right" | 1 || align="right" | 6 |
| align="right" | 1 || align="right" | 6 |
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| align="right" | 15 |
| align="right" | 15 |
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Vrstica 253: | Vrstica 253: | ||
|----- |
|----- |
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| align="center" | [[sedemkotniško število|Sedemkotniško]] |
| align="center" | [[sedemkotniško število|Sedemkotniško]] |
||
| align="center" | ½ |
| align="center" | ½''n''(5''n'' - 3) |
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| align="right" | 1 || align="right" | 7 |
| align="right" | 1 || align="right" | 7 |
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| align="right" | 18 |
| align="right" | 18 |
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Vrstica 264: | Vrstica 264: | ||
|----- |
|----- |
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| align="center" | [[osemkotniško število|Osemkotniško]] |
| align="center" | [[osemkotniško število|Osemkotniško]] |
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| align="center" | ½ |
| align="center" | ½''n''(6''n'' - 4) |
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| align="right" | 1 || align="right" | 8 |
| align="right" | 1 || align="right" | 8 |
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| align="right" | 21 |
| align="right" | 21 |
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Vrstica 275: | Vrstica 275: | ||
|----- |
|----- |
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| align="center" | [[devtkotniško število|Devetkotniško]] |
| align="center" | [[devtkotniško število|Devetkotniško]] |
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| align="center" | ½ |
| align="center" | ½''n''(7''n'' - 5) |
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| align="right" | 1 || align="right" | 9 |
| align="right" | 1 || align="right" | 9 |
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| align="right" | 24 |
| align="right" | 24 |
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Vrstica 286: | Vrstica 286: | ||
|----- |
|----- |
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| align="center" | [[desetkotniško število|Desetkotniško]] |
| align="center" | [[desetkotniško število|Desetkotniško]] |
||
| align="center" | ½ |
| align="center" | ½''n''(8''n'' - 6) |
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| align="right" | 1 || align="right" | 10 |
| align="right" | 1 || align="right" | 10 |
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| align="right" | 27 |
| align="right" | 27 |
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Vrstica 297: | Vrstica 297: | ||
|----- |
|----- |
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| align="center" | 11-kotniško |
| align="center" | 11-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(9''n'' - 7) |
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| align="right" | 1 || align="right" | 11 |
| align="right" | 1 || align="right" | 11 |
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| align="right" | 30 |
| align="right" | 30 |
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Vrstica 308: | Vrstica 308: | ||
|----- |
|----- |
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| align="center" | 12-kotniško |
| align="center" | 12-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(10''n'' - 8) |
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| align="right" | 1 || align="right" | 12 |
| align="right" | 1 || align="right" | 12 |
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| align="right" | 33 |
| align="right" | 33 |
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Vrstica 319: | Vrstica 319: | ||
|----- |
|----- |
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| align="center" | 13-kotniško |
| align="center" | 13-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(11''n'' - 9) |
||
| align="right" | 1 || align="right" | 13 |
| align="right" | 1 || align="right" | 13 |
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| align="right" | 36 |
| align="right" | 36 |
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Vrstica 330: | Vrstica 330: | ||
|----- |
|----- |
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| align="center" | 14-kotniško |
| align="center" | 14-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(12''n'' - 10) |
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| align="right" | 1 || align="right" | 14 |
| align="right" | 1 || align="right" | 14 |
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| align="right" | 39 |
| align="right" | 39 |
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Vrstica 341: | Vrstica 341: | ||
|----- |
|----- |
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| align="center" | 15-kotniško |
| align="center" | 15-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(13''n'' - 11) |
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| align="right" | 1 || align="right" | 15 |
| align="right" | 1 || align="right" | 15 |
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| align="right" | 42 |
| align="right" | 42 |
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Vrstica 352: | Vrstica 352: | ||
|----- |
|----- |
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| align="center" | 16-kotniško |
| align="center" | 16-kotniško |
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| align="center" | ½ |
| align="center" | ½''n''(14''n'' - 12) |
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| align="right" | 1 || align="right" | 16 |
| align="right" | 1 || align="right" | 16 |
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| align="right" | 45 |
| align="right" | 45 |
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Vrstica 363: | Vrstica 363: | ||
|----- |
|----- |
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| align="center" | 17-kotniško |
| align="center" | 17-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(15''n'' - 13) |
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| align="right" | 1 || align="right" | 17 |
| align="right" | 1 || align="right" | 17 |
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| align="right" | 48 |
| align="right" | 48 |
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Vrstica 374: | Vrstica 374: | ||
|----- |
|----- |
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| align="center" | 18-kotniško |
| align="center" | 18-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(16''n'' - 14) |
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| align="right" | 1 || align="right" | 18 |
| align="right" | 1 || align="right" | 18 |
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| align="right" | 51 |
| align="right" | 51 |
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Vrstica 385: | Vrstica 385: | ||
|----- |
|----- |
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| align="center" | 19-kotniško |
| align="center" | 19-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(17''n'' - 15) |
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| align="right" | 1 || align="right" | 19 |
| align="right" | 1 || align="right" | 19 |
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| align="right" | 54 |
| align="right" | 54 |
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Vrstica 396: | Vrstica 396: | ||
|----- |
|----- |
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| align="center" | 20-kotniško |
| align="center" | 20-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(18''n'' - 16) |
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| align="right" | 1 || align="right" | 20 |
| align="right" | 1 || align="right" | 20 |
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| align="right" | 57 |
| align="right" | 57 |
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Vrstica 407: | Vrstica 407: | ||
|----- |
|----- |
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| align="center" | 21-kotniško |
| align="center" | 21-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(19''n'' - 17) |
||
| align="right" | 1 || align="right" | 21 |
| align="right" | 1 || align="right" | 21 |
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| align="right" | 60 |
| align="right" | 60 |
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Vrstica 418: | Vrstica 418: | ||
|----- |
|----- |
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| align="center" | 22-kotniško |
| align="center" | 22-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(20''n'' - 18) |
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| align="right" | 1 || align="right" | 22 |
| align="right" | 1 || align="right" | 22 |
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| align="right" | 63 |
| align="right" | 63 |
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Vrstica 429: | Vrstica 429: | ||
|----- |
|----- |
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| align="center" | 23-kotniško |
| align="center" | 23-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(21''n'' - 19) |
||
| align="right" | 1 || align="right" | 23 |
| align="right" | 1 || align="right" | 23 |
||
| align="right" | 66 |
| align="right" | 66 |
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Vrstica 440: | Vrstica 440: | ||
|----- |
|----- |
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| align="center" | 24-kotniško |
| align="center" | 24-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(22''n'' - 20) |
||
| align="right" | 1 || align="right" | 24 |
| align="right" | 1 || align="right" | 24 |
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| align="right" | 69 |
| align="right" | 69 |
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Vrstica 451: | Vrstica 451: | ||
|----- |
|----- |
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| align="center" | 25-kotniško |
| align="center" | 25-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(23''n'' - 21) |
||
| align="right" | 1 || align="right" | 25 |
| align="right" | 1 || align="right" | 25 |
||
| align="right" | 72 |
| align="right" | 72 |
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Vrstica 462: | Vrstica 462: | ||
|----- |
|----- |
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| align="center" | 26-kotniško |
| align="center" | 26-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(24''n'' - 22) |
||
| align="right" | 1 || align="right" | 26 |
| align="right" | 1 || align="right" | 26 |
||
| align="right" | 75 |
| align="right" | 75 |
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Vrstica 473: | Vrstica 473: | ||
|----- |
|----- |
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| align="center" | 27-kotniško |
| align="center" | 27-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(25''n'' - 23) |
||
| align="right" | 1 || align="right" | 27 |
| align="right" | 1 || align="right" | 27 |
||
| align="right" | 78 |
| align="right" | 78 |
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Vrstica 484: | Vrstica 484: | ||
|----- |
|----- |
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| align="center" | 28-kotniško |
| align="center" | 28-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(26''n'' - 24) |
||
| align="right" | 1 || align="right" | 28 |
| align="right" | 1 || align="right" | 28 |
||
| align="right" | 81 |
| align="right" | 81 |
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Vrstica 495: | Vrstica 495: | ||
|----- |
|----- |
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| align="center" | 29-kotniško |
| align="center" | 29-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(27''n'' - 25) |
||
| align="right" | 1 || align="right" | 29 |
| align="right" | 1 || align="right" | 29 |
||
| align="right" | 84 |
| align="right" | 84 |
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Vrstica 506: | Vrstica 506: | ||
|----- |
|----- |
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| align="center" | 30-kotniško |
| align="center" | 30-kotniško |
||
| align="center" | ½ |
| align="center" | ½''n''(28''n'' - 26) |
||
| align="right" | 1 || align="right" | 30 |
| align="right" | 1 || align="right" | 30 |
||
| align="right" | 87 |
| align="right" | 87 |
Redakcija: 18:53, 14. marec 2008
Mnogokótniško števílo je v matematiki število, ki ga lahko razmestimo v obliki pravilnega mnogokotnika. Antični matematiki so ugotovili, da lahko števila razvrstimo na posebne načine, če si jih zamislimo kot kamenčke ali semena. Na primer število 10 lahko razmestimo v obliko trikotnika (glej trikotniško število):
x x x x x x x x x x
Števila 10 ne moremo postaviti v obliki kvadrata. Na drugi strani lahko tako postavimo število 9 (glej kvadratno število):
x x x x x x x x x
Nekatera števila, kot na primer 36, lahko razporedimo kot kvadrat ali kot trikotnik (glej trikotniško kvadratno število):
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
Neko mnogokotniško število povečamo s tem, da razširimo dve sosednji roki za eno točko in med točkami dodamo željeno število strani. Na naslednjih skicah, dodatni nivo prikažemo z znakom +.
Trikotniška števila
1:
+ x
3:
x x + + x x
6:
x x x x x x + + + x x x
10:
x x x x x x x x x x x x + + + + x x x x
Kvadratna števila
1:
+ x
4:
x + x x + + x x
9:
x x + x x x x x + x x x + + + x x x
16:
x x x + x x x x x x x + x x x x x x x + x x x x + + + + x x x x
Tudi mnogokotnike z večjim številom stranic, kot so petkotniki ali šestkotniki lahko predstavimo kot razvrstitve točk. Po dogovoru je 1 prvo mnogokotniško število za poljubno število stranic:
Petkotniška števila
1:
+ x
5:
x x + + x x + + x x
12:
x x x x x x + x x + x x x x + + x x + + + x x x
22:
x x x x x x x x x x x x x x + x x + x x x x + x x x + x x x x x + + x x + + + + x x x x
35:
x x x x x x x x x x x x x x x x x x x x x x + x x x x x + x x x x x x x + x x + x x x x + x x x x + x x x x x x + + x x + + + + + x x x x x
Šestkotniška števila
1:
x
6:
x x + + x x + + x x + x
15:
x x x x x x + x x + x x x x + x + x x x + + x x + + x x + x
28:
x x x x x x x x x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
45:
x x x x x x x x x x x x x x x x x x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
66:
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x + x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
91:
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + x x x x x x x + x x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x + x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
Če je s število stranic mnogokotnika, je enačba za n-to s-mnogokotniško število (n/2) ((s-2)n - (4-s)).
Ime | Enačba | n=1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
Trikotniško | ½n(1n + 1) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 |
Kvadratno | ½n(2n - 0) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 |
Petkotniško | ½n(3n - 1) | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 176 | 210 | 247 |
Šestkotniško | ½n(4n - 2) | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 231 | 276 | 325 |
Sedemkotniško | ½n(5n - 3) | 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | 286 | 342 | 403 |
Osemkotniško | ½n(6n - 4) | 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 341 | 408 | 481 |
Devetkotniško | ½n(7n - 5) | 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | 396 | 474 | 559 |
Desetkotniško | ½n(8n - 6) | 1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | 451 | 540 | 637 |
11-kotniško | ½n(9n - 7) | 1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | 506 | 606 | 715 |
12-kotniško | ½n(10n - 8) | 1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | 561 | 672 | 793 |
13-kotniško | ½n(11n - 9) | 1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | 616 | 738 | 871 |
14-kotniško | ½n(12n - 10) | 1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 671 | 804 | 949 |
15-kotniško | ½n(13n - 11) | 1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | 726 | 870 | 1027 |
16-kotniško | ½n(14n - 12) | 1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | 781 | 936 | 1105 |
17-kotniško | ½n(15n - 13) | 1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | 836 | 1002 | 1183 |
18-kotniško | ½n(16n - 14) | 1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 891 | 1068 | 1261 |
19-kotniško | ½n(17n - 15) | 1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | 946 | 1134 | 1339 |
20-kotniško | ½n(18n - 16) | 1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | 1001 | 1200 | 1417 |
21-kotniško | ½n(19n - 17) | 1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | 1056 | 1266 | 1495 |
22-kotniško | ½n(20n - 18) | 1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | 1111 | 1332 | 1573 |
23-kotniško | ½n(21n - 19) | 1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | 1166 | 1398 | 1651 |
24-kotniško | ½n(22n - 20) | 1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | 1221 | 1464 | 1729 |
25-kotniško | ½n(23n - 21) | 1 | 25 | 72 | 142 | 235 | 351 | 490 | 652 | 837 | 1045 | 1276 | 1530 | 1807 |
26-kotniško | ½n(24n - 22) | 1 | 26 | 75 | 148 | 245 | 366 | 511 | 680 | 873 | 1090 | 1331 | 1596 | 1885 |
27-kotniško | ½n(25n - 23) | 1 | 27 | 78 | 154 | 255 | 381 | 532 | 708 | 909 | 1135 | 1386 | 1662 | 1963 |
28-kotniško | ½n(26n - 24) | 1 | 28 | 81 | 160 | 265 | 396 | 553 | 736 | 945 | 1180 | 1441 | 1728 | 2041 |
29-kotniško | ½n(27n - 25) | 1 | 29 | 84 | 166 | 275 | 411 | 574 | 764 | 981 | 1225 | 1496 | 1794 | 2119 |
30-kotniško | ½n(28n - 26) | 1 | 30 | 87 | 172 | 285 | 426 | 595 | 792 | 1017 | 1270 | 1551 | 1860 | 2197 |
Glej tudi
Vir
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers, (Penguin Books, 1997) [ISBN 0140261494].
Zunanje povezave
- Mnogokotniška števila (MathWorld): http://mathworld.wolfram.com/PolygonalNumber.html (angleško)