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:

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 n^th triangular number i.e. T_n² = 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.

• There do not exist four distinct triangular numbers in arithmetic progression.

• There do not exist four distinct triangular numbers in geometric progression.

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

  T₁² + T₂² = 1 ² + 3 ²= 10 = T₄
   T₂² + T₃² = 3 ² + 6 ²= 45 = T₉
    T₃² + T₄² = 6 ² + 10 ²= 136 = T₁₆
     T_n-1² + T_n² = 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).

• 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 ?.

  (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.

  The largest known palindromic triangular number containing only odd digits:
  T_32850970 = 539593131395935

  The largest known palindromic triangular number containing only even digits:
  T_128127032 = 8208268228628028

  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.:

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.

• There exist infinite triangular numbers that are simultaneously the sum, the difference and the product of two triangular numbers, for example:

T₅₅ = T₁₀ + T₅₄ = T₁₅₄₀ - T₁₅₃₉ = T₇ * T₁₀

T₇₅ = T₂₉ + T₆₉ = T₇₇ - T₁₇ = T₅ * T₁₉

     and so on.

Some New Observations on Triangular Numbers :

• There are some triangular numbers which are product of three consecutive numbers. For example 210 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 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 > 1 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

  Note: The product of Ta*Tb is a perfect square for the formula (2a + 1)² - 1 = b. This produces a perfect square for EVERY triangular index and therefore every triangular number.

  (Note submitted by Steve Homewood vide his email dated 27 June 2018).

• 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.

  Observation: If n⁴ = T_a + T_b = T_c + T_d then a = n² - 1, b = n², c = a - n or n² - n - 1 and d = a + b - c - 1 or n² + n - 1.

  (Observation submitted by Steve Homewood vide his email dated 1st July 2018).

• 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
  An extension of above pattern:
  
  9⁰ + 9¹ + 9² + 9³ + 9⁴ = T₁₂₁ = T_{3⁰ + 3¹ + 3² + 3³ + 3⁴}

  (reported by Tony Foster vide his email dated 21 January 2024).

• 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
  (2*a + 1)² * T_n + T_a = T_{(2*a + 1)* n + a}
  
  For more on such identities refer Terry Trotter [11].

• 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

       

• The only known 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

  The only known examples of a Pythagorean triangle such that both Perimeter as well as Area are triangular numbers are:

  (3312, 14091, 14475) with Perimeter = 31878 = T₂₅₂ and Area = 23334696 = T₆₈₃₁

  (3405996, 8013265, 8707079) with Perimeter = 20126340 = T₆₃₄₄ and Area = 13646574268470 = T_5224284

  Can you find other examples ? For more visit Carlos Rivera.

• John McMahon has discovered the amazing curiosity [15] in the following Pythagorean triplets which are all generated by the Pythagorean triplet (5, 12, 13) by prefixing triangular numbers:
  15, 112, 113
  25, 312, 313
  35, 612, 613
  45, 1012, 1013
  55, 1512, 1513 and so on.
  This can be generalised as follows (where 5 is concatenated with n and, 12, 13 are concatenated with n^th triangular numbers):
  [n5]² + [T_n12]² = [T_n13]².

  (reported by Julian Beauchamp vide his email dated 24 November 2022).

• Sum of the consecutive Triangular Numbers:
  
  T₂₃² = T₃₃ + T₃₄ + T₃₅ + ..... + T₇₆ + T₇₇ + T₇₈

  T₂₃³ = T₉₃₃ + T₉₃₄ + T₉₃₅ + ..... + T₉₇₆ + T₉₇₇ + T₉₇₈

  (submitted by John McMahon vide his email dated 24 August 2020).

• Digits of Triangular Numbers
  
  In base 10, when the digit-length of T_k, the k-th triangular number, increases by 1 over the digit-length of T_k-1, the digits of k approximate the square root of either 2 (√ 2 = 1.4142135623730950488...) or 20 (√ 20 = 4.4721359549995793928...), depending on whether T_k has an odd or even number of digits. Here are some examples:

  T₄₅ = 1035 and 45² = 2025

  T₁₄₁ = 10011 and 141² = 19881

  T₄₄₇ = 100128 and 447² = 199809

  T₁₄₁₄ = 1000405 and 1414² = 1999396

  T₄₄₇₂ = 10001628 and 4472² = 19998784

  T₁₄₁₄₂ = 100005153 and 14142² = 199996164

  T₄₄₇₂₁ = 1000006281 and 44721² = 1999967841

  T₁₄₁₄₂₁ = 10000020331 and 141421² = 19999899241

  T₄₄₇₂₁₄ = 100000404505 and 447214² = 200000361796

  T_1414214 = 1000001326005 and 1414214 ² = 2000001237796

  T_4472136 = 10000002437316 and 4472136² = 20000000402496

  T_14142136 = 100000012392316 and 14142136² = 200000010642496

  T_44721360 = 1000000042485480 and 44721360² = 2000000040249600

  T_141421356 = 10000000037150046 and 141421356² = 19999999932878736

     and so on.

  In other bases, the approximated digits for k are of sqrt(2) and sqrt(2*base). In general, the k of p-sided polygonal numbers increasing by one digit approximates the square roots of 2p-4 and, under certain conditions, (2p-4)*base.

  (submitted by Mr Tony Sand vide his email dated 11 Mar 2022).

  Units Digit of Triangular numbers in base 10:
  

  The units digit of triangular numbers 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, etc.
  form a repeating sequence with two halves, where the second half is the mirror image of the first half as shown below:
  

  0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, ... (https://oeis.org/A008954 at the Online Encyclopedia of Integer Sequences)
  

  It can be seen that the sequence begins with 0, 1, 3, 6... and ends with ...6, 3, 1, 0.
  What explains this mirroring? Consider the second 5 in the sequence, which corresponds to T₉ = 45.
  

  45 + 10 = 55
  

  So the next digit in the sequence is 5, mirroring 5. Next consider the second 6 in the sequence, which corresponds to T₈ = 36 and so on.
  

  45 + 10 = 55
  36 + 9+10+11 = 66
  28 + 8+9+10+11+12 = 78
  21 + 7+8+9+10+11+12+13 = 91
  15 + 6+7+8+9+10+11+12+13+14 = 105
  10 + 5+6+7+8+9+10+11+12+13+14+15 = 120
  06 + 4+5+6+7+8+9+10+11+12+13+14+15+16 = 136
  03 + 3+4+5+6+7+8+9+10+11+12+13+14+15+16+17 = 153
  01 + 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18 = 171
  00 + 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19 = 190
  

  Note how the sum added to each triangular number is always a multiple of 10:
  45 + 10 = 45
  36 + 9+10+11 = 66 = 36 + (9+11)+10 = 36 + 20+10 = 36 + 30
  28 + 8+9+10+11+12 = 78 = 28 + (8+12)+(9+11)+10 = 28 + 20+20+10 = 28 + 50
  

  Adding a multiple of 10 leaves the units digit unaltered. And that's why the first half of the sequence is mirrored in the second half.
  

  The same is true if we consider the units digit of the triangular numbers in base 11.
  

  The units digit of triangular numbers (base 11) 0, 1, 3, 6, A, 14, 1A, 26, 33, 41, 50, 60, 71, 83, 96, AA, 114, 12A, 146, 163, 181, 1A0, 210...
  form a repeating sequence which is shorter and is mirrored in a slightly different way as shown below:
  

  0, 1, 3, 6, A, 4, A, 6, 3, 1, 0, 0, 1, 3, 6, A, 4, A, 6, 3, 1, 0, 0
  

  Here's why this happens:
  A + 5+6 = 1A (because 5+6 = 10 in base 11)
  6 + 4+5+6+7 = 26
  3 + 3+4+5+6+7+8 = 33
  1 + 2+3+4+5+6+7+8+9 = 41
  0 + 1+2+3+4+5+6+7+8+9+A = 50
  Note how the sum added to each triangular number is always a multiple of 11:
  

  A + 5+6 = 1A = A + 10
  6 + 4+5+6+7 = 26 = 6 + (4+7)+(5+6) = 6 + 10+10 = 6 + 20
  3 + 3+4+5+6+7+8 = 33 = 3 + (3+8)+(4+7)+(5+6) = 3 + 10+10+10 = 3 + 30
  

  Adding a multiple of 11 leaves the units digit unaltered in base 11.
  Base 10 provides the template for other even bases and base 11 provides the template for other odd bases.
  

  (submitted by Mr Tony Sand vide his email dated 12 May 2023).