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 = 1
T₂ = 1 + 2 = 3
T₃ = 1 + 2 + 3 = 6
T₄ = 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:

• The sum of two consecutive triangular numbers is always a square:

• If T is a Triangular number than 9*T + 1 is also a Triangular number:

• 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:

• 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 n^th triangular number i.e. T_n² = 1³ + 2³ + 3³ + ... + n³

• A triangular number greater than 1, can never be a Cube, a Fourth Power or a Fifth power.

• The sum of the squares of two consecutive triangular numbers is also a triangular number.

(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).

• All perfect numbers are triangular numbers

• The square of triangular numbers 1 and 6 produce triangular numbers 1 and 36.

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).

• 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 of reversible 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 n^th Square Triangular number K_n can easily be obtained from the recursive formula

  K_n = 34 * K_n-1 - K_n-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 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.

• 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.:

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.

• There are infinite 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 = T_4684659 

  19306982 * 19306983 = 372759573255306 = T_27304196  etc.

• 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 M^th 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 T_a and T_b (where T_a > 1 and T_b > and T_a is not equal to T_b) such that their product T_a*T_b 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 4^th 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 :

  T_n² = 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

• Triangular numbers appear in Pascal's Triangle. In fact 3rd diagonal of Pascal's Triangle, gives all triangular numbers as shown below:

       

  1
  1 1
  1 2 1
  1 3 3 1
  1 4 6 4 1
  1 5 10 10 5 1
  1 6 15 20 15 6 1
  1 7 21 35 35 21 7 1
  1 8 28 56 70 56 28 8 1
  1 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 = T_500+500 = 500500

  T₁₀₀₀₀ = T_5000+5000 = 50005000

  T₁₀₀₀₀₀ = T_50000+50000 = 5000050000

  T_1000000 = T_{500000+500000} = 500000500000     and so on.

• T_T5 + T_T6 = T₂₆

  T_T12 + T_T14 = T₆₁

  T_T77 + T_T89 = T₃₇₆

  T_T174 + T_T201 = T₈₇₁

  T_T1079 + T_T1249 = T₅₃₉₆

  T_T2430 + T_T2806 = 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 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.

  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.

• Sequence of sqrt(T) which are whole numbers is:
  
  √ T₁, √ T₈, √ T₄₉, √ T₂₈₈, √ T₁₆₈₁, √ T₉₈₀₀, √ T₅₇₁₂₁,..

  The product of any two consecutive numbers in above sequence is a triangular number, which is the product of two consecutive numbers:

  √ T₁ * √ T₈ = 6 = T₃ = 2 * 3

  √ T₈ * √ T₄₉ = 210 = T₂₀ = 14 * 15

  √ T₄₉ * √ T₂₈₈ = 7140 = T₁₁₉ = 84 * 85

  √ T₂₈₈ * √ T₁₆₈₁ = 242556 = T₆₉₆ = 492 * 493

  √ T₁₆₈₁ * √ T₉₈₀₀ = 8239770 = T₄₀₅₉ = 2870 * 2871

  √ T₉₈₀₀ * √ T₅₇₁₂₁ = 279909630 = T₂₃₆₆₀= 16730 * 16731 etc.

(submitted by Dr. N Mander (Mr) from Denmark vide his email dated 22 Oct 2011).

• There are infinite number of pairs (a,b) of positive integers such that 2 * T_a=T_b where a and b are integers and T_n is the n^th triangular number.

  2 * T₂ = T₃ = 6

  2 * T₁₄ = T₂₀ = 210

  2 * T₈₄ = T₁₁₉ = 7140

  2 * T₄₉₂ = T₆₉₆ = 242556

  2 * T₂₈₇₀ = T₄₀₅₉ = 8239770

  2 * T₁₆₇₃₀ = T₂₃₆₆₀ = 279909630

  2 * T₉₇₅₁₂ = T₁₃₇₉₀₃ = 9508687656

  2 * T₅₆₈₃₄₄ = T₈₀₃₇₆₀ = 323015470680

  2 * T_3312554 = T_4684659 = 10973017315470

  2 * T_19306982 = T_27304196 = 372759573255306   etc.

  With a₀ = 0 and b₀ = 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 ?.

References:

[1] Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

[2] Beiler, Albert H. Recreations in the Theory of Numbers. New York: Dover, 1966.

[3] Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.

[4] Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

[5] Gupta, Shyam Sunder "Curious Properties of triangular numbers." Science Reporter , September 1987, India.

[6] Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138 and 147-150, 1994.

[7] Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

[8] Madachy, Joseph S. Madachy's Mathematical Recreations. New York: Dover, 1979.

[9] Pickover, Clifford A. Wonders of Numbers. New York: Oxford University Press, 2001.

[10] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

[11] Trotter, T. Jr. "Some Identities for the Triangular Numbers." J. Recr. Math. 6, 128-135, 1973. http://www.trottermath.net/numthry/trident.html.

[12] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[13] Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin Books, 1991.

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