Fascinating Triangular Numbers |
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
T_{1} = 1
T_{2} = 1 + 2 = 3
T_{3} = 1 + 2 + 3 = 6
T_{4} = 1 + 2 + 3 + 4 = 10
So the n^{th} triangular number can be obtained as T_{n} = 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 n^{th} triangular number."
Some of the interesting properties of triangular numbers published in [5] are:
Curious properties of Triangular Numbers:
T_{1} + T_{2} = 1 + 3 = 4 = 2^{2}
T_{2} + T_{3} = 3 + 6 = 9 = 3^{2}
9*T_{1} + 1 = 9 * 1 + 1 = 10 = T_{4}
9*T_{2} + 1 = 9 * 3 + 1 = 28 = T_{7}
8*T_{1} + 1 = 8 * 1 + 1 = 9 = 3^{2}
8*T_{2} + 1 = 8 * 3 + 1 = 25 = 5^{2}
T_{4}^{2} = 10^{2} = 100 = 1^{3} + 2^{3} + 3^{3} + 4^{3}
T_{5}^{2} = 15^{2} = 225 = 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3}
T_{1}^{2} + T_{2}^{2} = 1 ^{2} + 3 ^{2}= 10 = T_{4}
T_{2}^{2} + T_{3}^{2} = 3 ^{2} + 6 ^{2}= 45 = T_{9}
T_{3}^{2} + T_{4}^{2} = 6 ^{2} + 10 ^{2}= 136 = T_{16}
T_{n-1}^{2} + T_{n}^{2} = T_{n2}
(Dr. Diego Marques, University of Brasilia, Brazil submitted vide his email dated 20 Apr 2011 that "this is also valid for the amazing Fibonacci sequence i.e. The sum of the squares of two consecutive fibonacci numbers is also a fibonacci number).
T_{1}^{2} = 1 * 1 = 1 = T_{1}
T_{3}^{2} = 6 * 6 = 36 = T_{8}
Can anybody find the third triangular number whose square is also a triangular number ?.
(AMRIK S NIMBRAN from Patna, Bihar submitted vide his email dated 21 Dec 2011 that "It took me some time to locate the proof of the impossibility of any other square triangular number beside 1 and 36 which is square of a triangular number, The source of proof is: L. J. Mordel, Diophantine Equations, 1969, Academic Press, London, Theorem 7, pp. 268-269).
K_{n} = 34 * K_{n-1} - K_{n-2} + 2.
So knowing the first two ST numbers i.e. K_{1} = 1 and K_{2} = 36 , all other successive Square Triangular numbers can be obtained , e.g.
K_{3} = 34 * K_{2} - K_{1} + 2 = 34 * 36 -1 + 2 = 1225
K_{4} = 34 * K_{3} - K_{2} + 2 = 34 * 1225 - 36 + 2 = 41616
The following non- recursive formula also gives n^{th} Square Triangular number in terms of variable n.
K_{n} = [{(1 + 2^{½})^{2n} - (1 - 2^{½})^{2n}}/(4*2^{½})]^{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.
21 + 15 = 36 = T_{8} : 21 - 15 = 6 = T_{3}
171 + 105 = 276 = T_{23} : 171 - 105 = 66 = T_{11}
703 + 378 = 1081 = T_{46} : 703 - 378 = 325 = T_{25}
and so on.
Some New Observations on Triangular Numbers :
1 * 2 * 3 = 6 = T_{3}
4 * 5 * 6 = 120 = T_{15}
5 * 6 * 7 = 210 = T_{20}
9 * 10 * 11 = 990 = T_{44}
56 * 57 * 58 = 185136 = T_{608}
636 * 637 * 638 = 258474216 = T_{22736}
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.
2 * 3 = 6 = T_{3}
14 * 15 = 210 = T_{20}
84 * 85 = 7140 = T_{119}
492 * 493 = 242556 = T_{696}
2870 * 2871 = 8239770 = T_{4059}
16730 * 16731 = 279909630 = T_{23660}
97512 * 97513 = 9508687656 = T_{137903}
568344 * 568345 = 323015470680 = T_{803760}
3312554 * 3312555 = 10973017315470 = T_{4684659}
19306982 * 19306983 = 372759573255306 = T_{27304196} etc.
2 * 3 = 6 = T_{3}
3 * 5 = 15 = T_{5}
3 * 7 = 21 = T_{6}
5 * 11 = 55 = T_{10}
7 * 13 = 91 = T_{13}
11 * 23 = 253 = T_{22}
19 * 37 = 703 = T_{37}
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.
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.
T_{2} * T_{24} = 3 * 300 = 900 = 30^{2}
T_{2} * T_{242} = 3 * 29403 = 88209 = 297^{2}
T_{3} * T_{48} = 6 * 1176 = 7056 = 84^{2}
T_{6} * T_{168} = 21 * 14196 = 298116 = 546^{2}
T_{11} * T_{528} = 66 * 139656 = 9217296 = 3036^{2}
T_{12} * T_{624} = 78 * 195000 = 15210000 = 3900^{2}
2^{4} = 16 = T_{3} + T_{4} = T_{1} + T_{5}
3^{4} = 81 = T_{8} + T_{9} = T_{5} + T_{11}
4^{4} = 256 = T_{15} + T_{16} = T_{11} + T_{19}
5^{4} = 625 = T_{24} + T_{25} = T_{19} + T_{29}
6^{4} = 1296 = T_{35} + T_{36} = T_{29} + T_{41}
7^{4} = 2401 = T_{48} + T_{49} = T_{41} + T_{55}
Observe the patterns formed above.
1 = T_{1}
1 + 9 = T_{4}
1 + 9 + 9^{2} = T_{13}
1 + 9 + 9^{2} + 9^{3} = T_{40}
1 + 9 + 9^{2} + 9^{3} + 9^{4} = T_{121}
T_{1} + T_{2} + T_{3}= T_{4}
T_{5} + T_{6} + T_{7} + T_{8} = T_{9} + T_{10}
T_{11} + T_{12} + T_{13} + T_{14} + T_{15}= T_{16} + T_{17} + T_{18}
T_{19} + T_{20} + T_{21} + T_{22} + T_{23} + T_{24} = T_{25} + T_{26} + T_{27} + T_{28}
T_{n}^{2} = T_{n} + T_{n-1} * T_{n+1}
T_{n2-1} = 2*T_{n} * T_{n-1}
For more on such identities visit Terry Trotter
8778^{2} + 10296^{2} = 13530^{2}
(T_{132})^{2} + (T_{143})^{2} = (T_{164})^{2}
Can you find other examples ? For more visit Carlos Rivera.
8 + 35
+ 23 = 9
+ 21 + 36
T_{28} + T_{29} + T_{30} + ... + T_{34} = T_{83}
T_{118} + T_{119} + T_{120} + ... + T_{128} = T_{322}
T_{16} + T_{17} + T_{18} + T_{19} = 2*T_{25}
T_{103} + T_{104} + T_{105} + T_{106} = 2*T_{148}
T_{10} = T_{5+5} = 55
T_{100} = T_{50+50} = 5050
T_{1000} = T_{500+500} = 500500
T_{10000} = T_{5000+5000} = 50005000
T_{100000} = T_{50000+50000} = 5000050000
T_{1000000} = T_{500000+500000} = 500000500000 and so on.
T_{T12} + T_{T14} = T_{61}
T_{T77} + T_{T89} = T_{376}
T_{T174} + T_{T201} = T_{871}
T_{T1079} + T_{T1249} = T_{5396}
T_{T2430} + T_{T2806}
= T_{12151} and so on.
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*10^{13} 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.
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 10^{5} 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 10^{5} 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.
Take an example of a 10-digit Triangular number 1061444835. It can be seen that this triangular number is the sum of the 10614^{th} and 44835^{th} triangular numbers. So the sum of two triangular numbers is equal to the number formed from concatenation of index of these two triangular numbers.
T_{10614} + T_{44835} = 1061444835
Other examples are:
T_{90} + T_{415} = 90415
T_{585} + T_{910} = 585910
T_{120} + T_{1545} = 1201545
T_{150} + T_{1726} = 1501726
T_{244} + T_{2196} = 2442196
T_{700} + T_{3676} = 7003676
T_{769} + T_{3846} = 7693846
T_{1474} + T_{5226} = 14745226
T_{2829} + T_{6970} = 28296970
T_{3030} + T_{7171} = 30307171 and so on.
Can you observe beautiful pattern in last two examples.
The product of any two consecutive numbers in above sequence is a triangular number, which is the product of two consecutive numbers:
√ T_{1} * √ T_{8} = 6 = T_{3} = 2 * 3
√ T_{8} * √ T_{49} = 210 = T_{20} = 14 * 15
√ T_{49} * √ T_{288} = 7140 = T_{119} = 84 * 85
√ T_{288} * √ T_{1681} = 242556 = T_{696} = 492 * 493
√ T_{1681} * √ T_{9800} = 8239770 = T_{4059} = 2870 * 2871
√ T_{9800} * √ T_{57121} = 279909630 = T_{23660}= 16730 * 16731 etc.
(submitted by Dr. N Mander (Mr) from Denmark vide his email dated 22 Oct 2011).
2 * T_{2} = T_{3} = 6
2 * T_{14} = T_{20} = 210
2 * T_{84} = T_{119} = 7140
2 * T_{492} = T_{696} = 242556
2 * T_{2870} = T_{4059} = 8239770
2 * T_{16730} = T_{23660} = 279909630
2 * T_{97512} = T_{137903} = 9508687656
2 * T_{568344} = T_{803760} = 323015470680
2 * T_{3312554} = T_{4684659} = 10973017315470
2 * T_{19306982} = T_{27304196} = 372759573255306 etc.
With a_{0} = 0 and b_{0} = 0 , the following recursive equations can give further values of a and b:
a_{n}=3 * a_{n-1} + 2 * b_{n-1} + 2
b_{n}=4 * a_{n-1} + 3 * b_{n-1} + 3
(submitted by Mr Roy Blatchford vide his email dated 17 Mar 2011).
Can anybody find similar recursive equations for pairs (a,b) of positive integers such that 3 * T_{a}=T_{b} ?.
6 * ( T_{1} + T_{2} + T_{3}) + 4 = 4^{3}
6 * ( T_{1} + T_{2} + T_{3} + T_{4}) + 5 = 5^{3}
6 * ( T_{1} + T_{2} + T_{3} + T_{4} + T_{5}) + 6 = 6^{3}
6 * ( T_{1} + T_{2} + T_{3} + T_{4} + T_{5} + T_{6}) + 7 = 7^{3}
6 * ( T_{1} + T_{2} + T_{3} + T_{4} + T_{5} + ...... + T_{n-1}) + n = n^{3}
(submitted by Don Davis vide his email dated 20 June 2013).
The sum of reciprocals of squares of all the triangular numbers converges to 4*(π^{2} - 9)/3:
1 + 1/3^{2} + 1/6^{2} + 1/10^{2} + 1/15^{2} + 1/21^{2} + 1/28^{2} + 1/36^{2} + 1/45^{2} + ....= 4*(π^{2} - 9)/3
The sum of reciprocals of cubes of all the triangular numbers converges to 8*(10 - π^{2}):
1 + 1/3^{3} + 1/6^{3} + 1/10^{3} + 1/15^{3} + 1/21^{3} + 1/28^{3} + 1/36^{3} + 1/45^{3} + ....= 8*(10 - π^{2})
(submitted by Okoh Ufuoma Cyrus from Nigeria vide his email dated 26 April 2013).
If you find any new and interesting observation about triangular numbers, please email me.
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This page is created on 26 Oct, 2002.