Numere poligonale
Transcript of Numere poligonale
In matematica, un numar poligonal este un numar reprezentat prin puncte sau cerculete (pietricele) aranjat sub forma unui poligon regulat. Primul numar dintr-o asemenea serie este intotdeauna 1, sau un punct. Al doilea numar este egal cu numarul de vertecsi ai poligonului respectiv. Al treilea numar poligonial este obtinut prin extensia a doua laturi ai celui de-al doilea numar poligonial, completand cu vertecsi poligonul marit acolo unde este necesar. Al treilea numar poligonial este egal cu suma vertecsilor din figura rezultata. Aveti mai jos cateva exemple.
Dintre acestea enumeram serii de numere triunghiulare, patrate, pentagonale, hexagonale, heptagonale, si tot asa... Aveti aici o lista cu 30 asemenea serii de numere reprezentate sub forma unui poligon regulat cat si formulele de calcul aferente. Am sa vorbesc putin in cele
ce urmeaza doar despre numerele triunghiulare ca astfel sa nu fie prea mare postarea prezenta. In postarea viitoare vom vorbi despre numerele patratice.
Numere triunghiulare.
Numerele triunghiulare sunt formate din sume partiale ale seriei 1+2+3+4+5+6+...+n. Asa ca:
Triunghiul 1 = T1 = 1
Triunghiul 2 = T2 = 1 + 2 = 3
T3 = 1 + 2 + 3 = 6
T4 = 1 + 2 + 3 + 4 = 10. Si asa mai departe.
Asa ca, numarul n va fi obtinut folosind formula T(n) = n*(n+1)/2, unde n este un numar natural.
Pasarile zboara uneori in stoluri de forma triunghiulara. Si avioanele, atunci cand zboara in formatie, folosesc aceasta forma triunghiulara. Propietatiile acestei serii de numere au fost studiate prima data de matematicienii Grecii antice. In postarea „Sa ne jucam cu matematica” v-am prezentat o metoda inedita prin care Karl Gauss a calculat in cateva secunde suma numerelor de la 1 la 100. Ajungem la acelasi rezultat folosind formula de calcul de mai sus, adica:
T(100) = 100*(100+1)/2 = 100/2 * 101 = 50*101 = 5050.
Va prezint cateva propietati curioase ale numerelor triunghiulare:
Suma a doua numere triunghiulare consecutive va fi intotdeauna un numar patratic:
o T1 + T2 = 1 + 3 = 4 = 22 sau T2 + T3 = 3 + 6 = 9 = 32
Daca T este un numar triunghiular, atunci 9*T + 1 va fi de asemenea un numar triunghiular:
o 9*T1 + 1 = 9*1 + 1 = 10 = T4 sau 9*T2 + 1 = 9*3 + 1 = 28 = T7
Daca T este un numar triunghiular, atunci 8*T + 1 va fi un patrat perfect:
o 8*T1 + 1 = 8*1 + 1 = 9 = 32 sau 8*T2 + 1 = 8*3 + 1 = 25 = 52
Un numar triunghiular nu se poate termina niciodata in 2, 4, 7 sau 9;
Orice numar intreg poate fi obtinut adunand pana la maxim trei numere triunghiulare: 51 = 15 + 36 = T5 + T8. Aceasta proprietate a fost descoperita si demonstrata de Karl Gauss acum 200 de ani.
Suma cifrelor unui numar triunghiular pana cand ramane o singura cifra va fi intotdeauna 1, 3, 6 sau 9;
Suma a n numere ridicate la cub consecutive incepand cu 1 este egal cu patratul celui de-al n numar triunghiular: T(n)2 = 13 + 23 + ... + n3.
o T(4)2 = 102 = 100 = 13 + 23 + 33 + 43 sau T(5)2 = 152 = 13 + 23 + 33 + 43 + 53.
Toate numerele perfecte sunt numere triunghiulare;
O proprietate interesanta:
o T1 + T2 + T3 = T4
o T5 + T6 + T7 + T8 = T9 + T10
o T11 + T12 + T13 + T14 + T15 = T16 + T17 + T18
o T19 + T20 + T21 + T22 + T23 + T24 = T25 + T26 + T27 + T28
Daca un grup de n persoane se saluta intre ei, numarul total de saluturi va fi numarul triunghiular (n-1).
The numbers which can be arranged in a compact triangular pattern are termed as triangular numbers. The triangular numbers are formed by partial sum of the series 1+2+3+4+5+6+7......+n. So
T1 = 1T2 = 1 + 2 = 3T3 = 1 + 2 + 3 = 6T4 = 1 + 2 + 3 + 4 = 10
So the nth triangular number can be obtained as Tn = n*(n+1)/2, where n is any natural number.In other words triangular numbers form the series 1,3,6,10,15,21,28.....
Flocks of birds often fly in this triangular formation. Even several airplanes when flying together constitute this formation. The properties of such numbers were first studied by ancient Greek mathematicians, particularly the Pythagoreans.
Have you heard of the following famous story about the famous mathematician Carl F. Gauss.
" The teacher asked everyone in the class to find the sum of all the numbers from 1 to 100. To everybody's surprise, Gauss stood up with the answer 5050 immediately. The teacher asked him as to how it was done. Gauss explained that instead of adding all the numbers from 1 to 100, add first and last term i.e. 1 + 100 =101, then add second and second last term i.e. 2 + 99 =101 and so on. Every pair sum is 101 and their will
be 50 such pairs ( total 100 numbers in all to be added), so 101 * 50 = 5050 is the answer. So the sum of numbers from 1 to N is (N/2)*(N+1), where N/2 are the number of pairs and N+1 is sum of each pair. This the famous formula for nth triangular number."
Some of the interesting properties of triangular numbers published in [5] are:
Curious properties of Triangular Numbers:
The sum of two consecutive triangular numbers is always a square:
T1 + T2 = 1 + 3 = 4 = 22
T2 + T3 = 3 + 6 = 9 = 32
If T is a Triangular number than 9*T + 1 is also a Triangular number:
9*T1 + 1 = 9 * 1 + 1 = 10 = T4
9*T2 + 1 = 9 * 3 + 1 = 28 = T7
A Triangular number can never end in 2, 4, 7 or 9:
If T is a Triangular number than 8*T + 1 is always a perfect square:
8*T1 + 1 = 8 * 1 + 1 = 9 = 32
8*T2 + 1 = 8 * 3 + 1 = 25 = 52
The digital root (i.e. ultimate sum of digits until a single digit is obtained) of triangular numbers is always 1,3,6 or 9.
The sum of n consecutive cubes starting from 1 is equal to the square of nth triangular number i.e. Tn
2 = 13 + 23 + 33 + ... + n3
T42 = 102 = 100 = 13 + 23 + 33 + 43
T52 = 152 = 225 = 13 + 23 + 33 + 43 + 53
A triangular number greater than 1, can never be a Cube, a Fourth Power or a Fifth power.
All perfect numbers are triangular numbers
The square of triangular numbers 1 and 6 produce triangular numbers 1 and 36.
T12 = 1 * 1 = 1 = T1
T32 = 6 * 6 = 36 = T8
Can anybody find the third triangular number whose square is also a triangular number ?.
The numbers in the sequence 1, 11, 111, 1111, 11111,...etc. are all triangular numbers in base 9.
The only triangular number which is also a prime is 3.
Palindromic Triangular Numbers: Some of the many triangular numbers, which are also palindromic ( i.e. reading the same forward as well as backward) are 1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15051, 66066, 617716, 828828, 1269621, 1680861, 3544453, 5073705, 5676765, 6295926, 351335153, 61477416, 178727871, 1264114621, 1634004361 etc. These can be termed as palindromic triangular numbers. There are 28 Palindromic Triangular numbers below 1010. For more on these numbers visit Patrick De Geest
Reversible Triangular Numbers: Some of the many triangular numbers, whose reversals are also triangular numbers are 1,3,6,10,55,66,120,153,171,190,300,351,595,630,666,820,3003,5995,8778,15051, 17578,66066,87571,156520,180300,185745,547581,557040,617716,678030,828828, 1269621,1461195,1680861,1851850,3544453,5073705,5676765,5911641,6056940, 6295926,12145056,12517506,16678200,35133153,56440000,60571521,61477416, 65054121,157433640,178727871,188267310,304119453,354911403,1261250200, 1264114621,1382301910,1634004361,1775275491,1945725771 etc. These can be termed as Reversible triangular numbers. Note that all palindromic triangular numbers mentioned above are special case ofreversible triangular numbers.
Square Triangular Numbers: There are infinitely many triangular numbers, which are also squares as given by the series 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 etc. These can be termed as Square triangular(ST) numbers. The nth Square Triangular number Kn can easily be obtained from the recursive formula
Kn = 34 * Kn-1 - Kn-2 + 2.
So knowing the first two ST numbers i.e. K1 = 1 and K2 = 36 , all other successive Square Triangular numbers can be obtained , e.g.
K3 = 34 * K2 - K1 + 2 = 34 * 36 -1 + 2 = 1225
K4 = 34 * K3 - K2 + 2 = 34 * 1225 - 36 + 2 = 41616
The following non- recursive formula also gives nth Square Triangular number in terms of variable n.
Kn = [{(1 + 2½)2n - (1 - 2½)2n}/(4*2½)]2
It is interesting to note that digital root of all EVEN Square Triangular Numbers i.e. 36, 41616, 48024900, 55420693056 .. etc is always 9 and digital root of all ODD Square Triangular Numbers i.e. 1, 1225, 1413721, 1631432881, ... etc is always 1. Also Square Triangular Numbers can never end in 2, 3, 4, 7, 8 or 9.
There are pairs of triangular numbers such that the sum and difference of numbers in each pair are also triangular numbers e.g. (15, 21), (105, 171), (378, 703), (780, 990), (1485, 4186), (2145, 3741), (5460, 6786), (7875, 8778)... etc.:
21 + 15 = 36 = T8 : 21 - 15 = 6 = T3
171 + 105 = 276 = T23 : 171 - 105 = 66 = T11
703 + 378 = 1081 = T46 : 703 - 378 = 325 = T25
and so on.
Some New Observations on Triangular Numbers :
There are some triangular numbers which are product of three consecutive numbers. For example 120 is a triangular number which is product of three consecutive numbers 5, 6 and 7.There are only 6 such triangular numbers ,largest of which is 258474216, as shown below:
1 * 2 * 3 = 6 = T3
4 * 5 * 6 = 120 = T15
5 * 6 * 7 = 210 = T20
9 * 10 * 11 = 990 = T44
56 * 57 * 58 = 185136 = T608
636 * 637 * 638 = 258474216 = T22736
The triangular number 120 is the product of three, four and five consecutive numbers.
4 * 5 * 6 = 2 * 3 * 4 * 5 = 1 * 2 * 3 * 4 * 5 = 120
No other triangular number is known to be the product of four or more consecutive numbers.
There are some triangular numbers which are product of two consecutive numbers. For example 6 is a triangular number which is product of two consecutive numbers 2 and 3. Some others are as shown below:
2 * 3 = 6 = T3
14 * 15 = 210 = T20
84 * 85 = 7140 = T119
492 * 493 = 242556 = T696
2870 * 2871 = 8239770 = T4059
16730 * 16731 = 279909630 = T23660
97512 * 97513 = 9508687656 = T137903
568344 * 568345 = 323015470680 = T803760
3312554 * 3312555 = 10973017315470 = T4684659
19306982 * 19306983 = 372759573255306 = T27304196
The triangular numbers which are product of two prime numbers can be termed as Triangular Semiprimes . For example 6 is a Triangular Semiprimes . Some other examples of Triangular Semiprimes are:10,15,21,55,91,253,703,1081,1711,1891,2701,3403,5671,12403,13861,15931, 18721,25651,34453,38503,49141,60031,64261,73153,79003,88831,104653,108811, 114481,126253,146611,158203,171991,188191,218791,226801,258121,269011,286903, 351541,371953,385003,392941,482653,497503 etc as shown below:
2 * 3 = 6 = T3
3 * 5 = 15 = T5
3 * 7 = 21 = T6
5 * 11 = 55 = T10
7 * 13 = 91 = T13
11 * 23 = 253 = T22
19 * 37 = 703 = T37
Harshad Triangular Numbers:
Harshad (or Niven ) numbers are those numbers which are divisible by their sum of the digits. For example 1729 ( 19*91) is divisible by 1+7+2+9 =19, so 1729 is a Harshad number.
Harshad Triangular Number can be defined as the Triangular numbers which are divisible by the sum of their digits. For example, Triangular number 1128 is divisible by 1+1+2+8 = 12 (i.e. 1128/12 = 94). So 1128 is a Harshad Triangular Number. Other examples are:
1, 3, 6, 10, 21, 36, 45, 120, 153, 171, 190, 210, 300, 351, 378, 465, 630, 666, 780, 820, 990, 1035, 1128, 1275, 1431, 1540, 1596, 1770, 2016, 2080, 2556, 2628, 2850, 2926, 3160, 3240, 3321, 3486, 3570, 4005, 4465, 4560, 4950, 5050, 5460, 5565, 5778, 5886, 7140, 7260, 8001, 8911, 9180, 10011, 10296, 10440, 11175, 11476, 11628, 12720, 13041, 13203, 14196, 14706, 15225, 15400, 15576, 16110, 16290, 16653, 17020, 17205, 17766, 17955, 18145, 18528, 20100, 21321, 21528, 21736, 21945, 22155, 23220, 23436, 24090, 24310, 24976, 25200, 28680, 29646, 30628, 31626, 32640, 33930, 35245, 36585, 37128, 39060, 40470, 41328, 41616, 43365, 43956, 45150, 46360, 51040, 51360, 51681, 52326, 52650, 53956, 56280, 56616, 61776, 63903, 64620, 65341, 67896, 69006, 70125, 70500, 72010, 73536, 73920, 76636, 78210, 79401, 79800, 80200,81810, 88410, 89676, 90100, 93096, 93528, 97020, 100128, 101025, 103740, 105111, 105570 etc.
Happy Triangular Numbers:
If you iterate the process of summing the squares of the decimal digits of a number and if the process terminates in 1, then the original number is called a Happy number. For example 7 -> 49 -> 97 -> 130 -> 10 -> 1.
A Happy Triangular Number is defined as a Triangular number which is also a Happy number. For example, consider a triangular number 946, where 946 -> 133 -> 19 -> 82 -> 68 -> 100 -> 1. So 946 is a Happy triangular Number. Other examples of Happy Triangular Numbers are :
1, 10, 28, 91, 190, 496, 820, 946, 1128, 1275, 2080, 2211, 2485, 3321, 4278, 8128, 8256, 8778, 9591, 9730, 11476, 12090, 12880, 13203, 13366, 13530, 15753, 16471, 17205, 17578, 20910, 21115, 21321, 22791, 24753, 25651,
27261, 29890, 30135, 31626, 33670, 35245, 36046, 41328, 43660, 43956, 44253, 46360, 47586, 48205, 50721, 53301, 53956, 54615, 55278, 56280, 56953, 58311, 61425, 62128, 66430, 69378, 69751, 70125, 75855, 76245, 77815, 79003, 80200, 81810, 82621, 84666, 87571, 90100, 90951, 93961, 99681, 100128, 101025, 102831, 103285, 105570, 107416, 110215, 117370, 119316, 122760, 123256, 123753, 126253, 127260, 129286, 130305 etc.
The only Fibonacci Numbers that are also triangular are 1, 3, 21 and 55.
The only triangular Numbers that are also repdigit are 55, 66 and 666.
3 is the only triangular Number that is also a Fermat Number.
Every Hexagonal number is a triangular number.
Every pentagonal number is one-third of a triangular number.
If M is a Mersenne prime then Mth triangular number is a perfect number.
The sum of reciprocals of triangular numbers converges to 2:
1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28 + 1/36 + 1/45 + ....= 2
There are many pairs of triangular numbers Ta and Tb (where Ta > 1 and Tb > and Ta is not equal to Tb) such that their product Ta*Tb is a Perfect Square. For example:
T2 * T24 = 3 * 300 = 900 = 302
T2 * T242 = 3 * 29403 = 88209 = 2972
T3 * T48 = 6 * 1176 = 7056 = 842
T6 * T168 = 21 * 14196 = 298116 = 5462
T11 * T528 = 66 * 139656 = 9217296 = 30362
T12 * T624 = 78 * 195000 = 15210000 = 39002
Every positive integer can be written as a sum of atmost three triangular numbers.
Every 4th power greater than 1, is the sum of two triangular numbers.
24 = 16 = T3 + T4 = T1 + T5
34 = 81 = T8 + T9 = T5 + T11
44 = 256 = T15 + T16 = T11 + T19
54 = 625 = T24 + T25 = T19 + T29
64 = 1296 = T35 + T36 = T29 + T41
74 = 2401 = T48 + T49 = T41 + T55
Observe the patterns formed above.
For any natural number n, the number 1 + 9 + 92 + 93 + ... + 9n is a triangular number.
1 = T1
1 + 9 = T4
1 + 9 + 92 = T13
1 + 9 + 92 + 93 = T40
1 + 9 + 92 + 93 + 94 = T121
A curious pattern :
T1 + T2 + T3= T4
T5 + T6 + T7 + T8 = T9 + T10
T11 + T12 + T13 + T14 + T15= T16 + T17 + T18
T19 + T20 + T21 + T22 + T23 + T24 = T25 + T26 + T27 + T28
Some identities :
Tn2 = Tn + Tn-1 * Tn+1
Tn2
-1 = 2*Tn * Tn-1
For more on such identities visit Terry Trotter
Triangular numbers appear in Pascal's Triangle. In fact 3rd diagonal of Pascal's Triangle, gives all triangular numbers as shown below:
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1
One example of a Pythagorean triangle (a,b,c) where a, b, c are triangular numbers is (8778, 10296, 13530)
87782 + 102962 = 135302
(T132)2 + (T143)2 = (T164)2
Can you find other examples ? For more visit Carlos Rivera.
T8 + T35 + T23 = T9 + T21 + T36
8 + 35 + 23 = 9 + 21 + 36
Sum of Consecutive Triangular Numbers:
T28 + T29 + T30 + ... + T34 = T83
T118 + T119 + T120 + ... + T128 = T322
T16 + T17 + T18 + T19 = 2*T25
T103 + T104 + T105 + T106 = 2*T148
Observe this pattern of infinite triangular numbers.
T10 = T5+5 = 55
T100 = T50+50 = 5050
T1000 = T500+500 = 500500
T10000 = T5000+5000 = 50005000
T100000 = T50000+50000 = 5000050000
T1000000 = T500000+500000 = 500000500000 and so on.
TT5 + TT6 = T26
TT12 + TT14 = T61
TT77 + TT89 = T376
TT174 + TT201 = T871
TT1079 + TT1249 = T5396
TT2430 + TT2806 = T12151 and so on.
Highly Composite Triangular Numbers:
Numbers such that d(n), the number of divisors of n, is greater than for any smaller n are called highly composite numbers. If n is a triangular number then it can be termed as Highly Composite Triangular Number . For example 28 is a triangular number and d(28) = 6 . Number of divisors of all triangular numbers less than 28 is less than 6. So 28 is a Highly Composite Triangular number.
All Highly Composite Triangular numbers below 5*1013 are:
1, 3, 6, 28, 36, 120, 300, 528, 630, 2016, 3240, 5460, 25200, 73920, 157080, 437580, 749700, 1385280, 1493856, 2031120, 2162160, 17907120, 76576500, 103672800, 236215980, 842161320, 3090906000, 4819214400, 7589181600, 7966312200, 13674528000, 20366564400, 49172323200, 78091322400, 102774672000, 557736444720, 666365279580, 876785263200, 1787835551040, 2427046221600, 3798207594720, 24287658595200 and 26571463158240.
Abundant and Deficient Triangular Numbers:
Numbers such that s(n), the sum of aliquot divisors of n, is greater than n are called Abundant numbers. If n is a triangular number then it can be termed as Abundant Triangular Number . For example 36 is a triangular number and s(36) = 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55, which is greater than 36. So 36 is a Abundant Triangular number.
All Abundant Triangular numbers below 105 are:
36,66,78,120,210,276,300,378,528,630,666,780,820,990,1128,1176,1326,1540,1596, 1770,1830,2016,2080,2346,2556,2628,2850,3160,3240,3486,3570,3828,4095,4278, 4560,4656,4950,5460,5778,5886,6216,6328,6786,7140,7260,7626,7875,8256,8646, 8778,9180,9730,9870,10296,10440,10878,11628,12090,12246,12720,12880,13530, 14028,14196,14706,15400,15576,16110,16290,16836,17020,17766,17955,18336,18528, 19110,19900,20100,20706,20910,21528,21736,21945,22578,23220,23436,24090,24310, 24976,25200,25878,26106,26565,26796,27966,28680,28920,29646,29890,30628,30876, 31626,31878,32640,33670,33930,34716,34980,35778,37128,37950,38226,39060,39340, 40470,41328,41616,42486,43660,43956,44850,45150,46056,46360,47586,48516,48828, 49770,51040,51360,52326,52650,53628,53956,54285,55278,56280,56616,57630,57970, 58996,59340,60378,60726,61425,61776,62128,63546,64620,64980,66066,66430,67896, 69006,69378,70500,72390,73536,73920,75078,76636,77028,78210,78606,79800,80200, 81810,82215,83028,83436,84666,86320,86736,87990,88410,89676,90100,91806,93096, 93528,94830,96580,97020,98346 and 98790.
Numbers such that s(n), the sum of aliquot divisors of n, is less than n are called Deficient numbers. If n is a triangular number then it can be termed as Deficient Triangular Number . For example 21 is a triangular number and s(21) = 1 + 3 + 7 = 11, which is less than 21. So 21 is a Deficient Triangular number.
All Deficient Triangular numbers below 105 are:
1,3,10,15,21,45,55,91,105,136,153,171,190,231,253,325,351,406,435,465, 561,595,703,741,861,903,946,1035,1081,1225,1275,1378,1431,1485,1653,1711, 1891,1953,2145,2211,2278,2415,2485,2701,2775,2926,3003,3081,3321,3403,3655, 3741,3916,4005,4186,4371,4465,4753,4851,5050,5151,5253,5356,5565,5671,5995, 6105,6441,6555,6670,6903,7021,7381,7503,7750,8001,8385,8515,8911,9045, 9316,9453,9591,10011,10153,10585,10731,11026,11175,11325,11476,11781,
11935, 12403,12561,13041,13203,13366,13695,13861,14365,14535,14878,15051,15225,15753, 15931,16471,16653,17205,17391,17578,18145,18721,18915,19306,19503,19701,20301, 20503,21115,21321,22155,22366,22791,23005,23653,23871,24531,24753,25425,25651, 26335,27028,27261,27495,27730,28203,28441,29161,29403,30135,30381,31125,31375, 32131,32385,32896,33153,33411,34191,34453,35245,35511,36046,36315,36585,36856, 37401,37675,38503,38781,39621,39903,40186,40755,41041,41905,42195,42778,43071, 43365,44253,44551,45451,45753,46665,46971,47278,47895,48205,49141,49455,50086, 50403,50721,51681,52003,52975,53301,54615,54946,55611,55945,56953,57291,58311, 58653,59685,60031,61075,62481,62835,63190,63903,64261,65341,65703,66795,67161, 67528,68265,68635,69751,70125,70876,71253,71631,72010,72771,73153,74305,74691, 75466,75855,76245,77421,77815,79003,79401,80601,81003,81406,82621,83845,84255, 85078,85491,85905,87153,87571,88831,89253,90525,90951,91378,92235,92665,93961, 94395,95266,95703,96141,97461,97903,99235 and 99681.
Curious Sum of Two Triangular Numbers.
Take an example of a 10-digit Triangular number 1061444835. It can be seen that this triangular number is the sum of the 10614th and 44835th triangular numbers. So the sum of two triangular numbers is equal to the number formed from concatenation of index of these two triangular numbers.
T10614 + T44835 = 1061444835
Other examples are:
T90 + T415 = 90415
T585 + T910 = 585910
T120 + T1545 = 1201545
T150 + T1726 = 1501726
T244 + T2196 = 2442196
T700 + T3676 = 7003676
T769 + T3846 = 7693846
T1474 + T5226 = 14745226
T2829 + T6970 = 28296970
T3030 + T7171 = 30307171 and so on.
Can you observe beautiful pattern in last two examples.
Definition and examples
The number 10, for example, can be arranged as a triangle (see triangular number):
But 10 cannot be arranged as a square. The number 9, on the other hand, can
be (see square number):
Some numbers, like 36, can be arranged both as a square and as a triangle
(see square triangular number):
By convention, 1 is the first polygonal number for any number of sides.
The rule for enlarging the polygon to the next size is to extend two
adjacent arms by one point and to then add the required extra sides
between those points. In the following diagrams, each extra layer is
shown as in red.
Triangular numbers
Square numbers
Polygons with higher numbers of sides, such as pentagons and
hexagons, can also be constructed according to this rule, although the
dots will no longer form a perfectly regular lattice like above.
Pentagonal numbers
Hexagonal numbers
[edit]Formula
If s is the number of sides in a polygon, the formula for the nth s-gonal
number P(s,n) is
The nth s-gonal number is also related to the triangular numbers Tn as
follows:
Thus:
For a given s-gonal number P(s,n) = x, one can find n by
[edit]Table of values
s NameFormu
la
n
=12 3 4 5 6 7 8 9 10 Sum of Reciprocals[1]
3 Triangular½(1n²
+ 1n)1 3 6 10 15 21 28 36 45 55
4 Square½(2n²
- 0n)1 4 9 16 25 36 49 64 81 100
5 Pentagonal½(3n²
- 1n)1 5
1
222 35 51 70 92
11
7145
6 Hexagonal½(4n²
- 2n)1 6
1
528 45 66 91
12
0
15
3190
7 Heptagonal½(5n²
- 3n)1 7
1
834 55 81
11
2
14
8
18
9235
[2]
8 Octagonal½(6n²
- 4n)1 8
2
140 65 96
13
3
17
6
22
5280
9 Nonagonal½(7n²
- 5n)1 9
2
446 75
11
1
15
4
20
4
26
1325
1
0Decagonal
½(8n²
- 6n)1
1
0
2
752 85
12
6
17
5
23
2
29
7370
1
1
Hendecagon
al
½(9n²
- 7n)1
1
1
3
058 95
14
1
19
6
26
0
33
3415
1
2
Dodecagona
l
½(1
0n² -
8n)
11
2
3
364
10
5
15
6
21
7
28
8
36
9460
1
3Tridecagonal
½(1
1n² -
9n)
11
3
3
670
11
5
17
1
23
8
31
6
40
5505
1
4
Tetradecago
nal
½(1
2n² -
10n)
11
4
3
976
12
5
18
6
25
9
34
4
44
1550
1
5
Pentadecag
onal
½(1
3n² -
11n)
11
5
4
282
13
5
20
1
28
0
37
2
47
7595
1
6
Hexadecago
nal
½(1
4n² -
12n)
11
6
4
588
14
5
21
6
30
1
40
0
51
3640
1
7
Heptadecag
onal
½(1
5n² -
13n)
11
7
4
894
15
5
23
1
32
2
42
8
54
9685
1
8
Octadecago
nal
½(1
6n² -
14n)
11
8
5
1
10
0
16
5
24
6
34
3
45
6
58
5730
1
9
Nonadecago
nal
½(1
7n² -
15n)
11
9
5
4
10
6
17
5
26
1
36
4
48
4
62
1775
2
0Icosagonal
½(1
8n² -
16n)
12
0
5
7
11
2
18
5
27
6
38
5
51
2
65
7820
2
1
Icosihenago
nal
½(1
9n² -
17n)
12
1
6
0
11
8
19
5
29
1
40
6
54
0
69
3865
2
2Icosidigonal
½(2
0n² -
18n)
12
2
6
3
12
4
20
5
30
6
42
7
56
8
72
9910
2
3Icositrigonal
½(2
1n² -
19n)
12
3
6
6
13
0
21
5
32
1
44
8
59
6
76
5955
2
4
Icositetragon
al
½(2
2n² -
20n)
12
4
6
9
13
6
22
5
33
6
46
9
62
4
80
1
100
0
The On-Line Encyclopedia of Integer Sequences eschews terms using Greek
prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal").
Combinations
Some numbers, such as 36 which is both square and triangular, fall into two
polygonal sets. The problem of determining, given two such sets, all numbers
that belong to both can be solved by reducing the problem to Pell's equation.
The simplest example of this is the sequence of square triangular numbers.
The following table summarizes the set of s-gonal t-gonal numbers for small
values of s and t.
s t Sequence
4 3 1, 36, 1225, 41616, …
5 3 1, 210, 40755, 7906276, …
5 4 1, 9801, 94109401, …
6 3 All hexagonal numbers are also triangular.
6 4 1, 1225, 1413721, 1631432881, …
6 5 1, 40755, 1533776805, …
7 3 1, 55, 121771, 5720653, …
7 4 1, 81, 5929, 2307361, …
7 5 1, 4347, 16701685, 64167869935, …
7 6 1, 121771, 12625478965, …
8 3 1, 21, 11781, 203841, …
8 4 1, 225, 43681, 8473921, …
8 5 1, 176, 1575425, 234631320, …
8 6 1, 11781, 113123361, …
8 7 1, 297045, 69010153345, …
9 3 1, 325, 82621, 20985481, …
9 4 1, 9, 1089, 8281, 978121, …
9 5 1, 651, 180868051, …
9 6 1, 325, 5330229625, …
9 7 1, 26884, 542041975, …
9 8 1, 631125, 286703855361, …
In some cases, such as s=10 and t=4, there are no numbers in both sets other
than 1.
The problem of finding numbers that belong to three polygonal sets is more
difficult. A computer search for pentagonal square triangular numbers has yielded
only the trivial value of 1, though a proof that there are no such number has yet to
appear in print.[3] All hexagonal square numbers are also hexagonal square
triangular numbers, and 1225 is actually a hecticositetragonal, hexacontagonal,
icosinonagonal, hexagonal, square, triangular number.
Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
Number of legal ways to insert a pair of parentheses in a string of n letters. E.g. there are 6 ways for three letters: (a)bc, (ab)c, (abc), a(b)c, a(bc), ab(c).
The squares: a(n) = n^2.
Pentagonal numbers: n*(3*n-1)/2.
Hexagonal numbers: n*(2*n-1).
Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.
Octagonal numbers: n*(3*n-2). Also called star numbers.
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,1,... - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 21 2001. The spiral begins:
......16..15..14
....17..5...4...13
..18..6...0...3...1219..7...1...2...11..26..20..8...9...10..25....21..22..23..24a(n) = (3n-2)(3n-1)(3n)/{(3n-1)+(3n-2)+(3n)} i.e. the product of three consecutive numbers/their sum. a(1) = 1*2*3/(1+2+3),a(2) = 4*5*6/(4+5+6)
a(0) = 0, a(1) = 1; for n >= 2, a(n) = 34 * a(n-1) - a(n-2) + 2.