Properties of Triangular Numbers |
* Triangular No. are expressed by a general formula n(n+1) / 2 where n is an integer. Thus T (n) = n(n+1) / 2 . These numbers cannot have 2,4,7,9 as their last digits. |
* Square of an integer n can be expressed as a sum of Triangular no. T(n) & T(n-1). Thus n^{2 }= T(n) + T(n-1) |
* Some triangular numbers are squares. Exa- 1, 36, 1225,
41616,1413721, 48024900, 1631432881, 55420693056, 1882672131025 which are squares of 1,6, 35, 204,
1189, 6930, 40391, 235416, 1372105 respectively. Triangular numbers which are squares must have the last digit any of the 1,5,6,0 and satisfy the equation T(n)=n^{2} so that 8n^{2} + 1= x^{2} where n is an integer and x is also an integer. Putting 2n=y, we get x^{2} - 2 y^{2 }= 1 which is a Diophantine Equation. It is also a Pell's Equation. Pell's Equation is a Diophantine equation of the form x^{2} - n y^{2 }= 1 where n is a given non-square integer and integer solutions are sought for x & y. Square Triangular are given by the recursive formula -- K_{n} = 34 * K_{n-1} - K_{n-2} + 2. |
* T_{n}^{2} = 1^{3} + 2^{3} + 3^{3} + ... + n^{3} |
* 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. |
* The only triangular number which is also a prime is 3. |
* The sum of the digits of all EVEN & ODD square triangular numbers when reduced to single digit become 9 & 1 respectively. |
* 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 triangular numbers which are product of three consecutive numbers. There are 6 such triangular numbers. 6(1*2*3),120(4*5*6),210(5*6*7),990(9*10*11),185136(56*57*58),258474216(636*637*638) |
* 120 is the product of 3(4*5*6),4(2*3*4*5) & 5 (1*2*3*4*5) consecutive numbers. No other triangular number is found to be the product of 4 or more consecutive numbers. |
* The triangular numbers which are product of two prime numbers can be termed as Triangular Semi-primes . For example 6 is a Triangular Semi-primes . Some other examples of Triangular Semi-primes 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 |
* The only Fibonacci Numbers that are also triangular are 1, 3, 21 and 55. |
* The sum of reciprocals of triangular numbers converges to 2. 1/1 +1/3 +1/6 +1/10 + 1/15 + 1/21 +1/28 +1/36 + 1/45 + 1/55 +.......... =2. Huygens had suggested to Leibniz to determine the sum of the reciprocals of the so called triangular numbers. |
* For any natural number n, the number 1 + 9 + 9^{2} + 9^{3} + ... + 9^{n} is a triangular number. |
* One example of a Pythagorean triangle (a,b,c) where a, b, c are triangular numbers is (8778, 10296, 13530) |
* The sum of reciprocals of squares of all the triangular numbers converges to 4*(π^{2} - 9)/3 |
* The sum of reciprocals of cubes of all the triangular numbers converges to 8*(10 - π^{2}) |
* 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 10^{10} |
* Triangular numbers which are divisible by the sum of their digits are called Harshad numbers. Exa-- 1, 3, 6, 10, 21, 36, 45, 120, 153, 171, 190, 210, 300 etc |
* There are many pairs of triangular numbers such that their product is a perfect square. Exa-- T(2)*T(24) = 30*30 ; T(2)*T(242) = 297*297 ;T(3)*T(48) = 84*84 ; T(6)*T(168) = 546*546 ; T(11)*T(528) = 3036*3036 ; T(12)*T(624) = 3900*3900 ; |
* Every 4^{th} power greater than 1, is the sum of two triangular numbers. Exa- 2^{4} = T(1) + T(5) ; 3^{4} = T(5) + T(11) =T(8) + T(9) ; 4^{4} = T(11) + T(19) =T(15) + T(16); 5^{4} = T(19) + T(29) =T(24) + T(25) ; 6^{4} = T(29) + T(41) =T(35) + T(36) ; 7^{4} = T(41) + T(55) =T(48) + T(49) ; |
* There are infinite no. of triangular numbers which are product of 2 consecutive numbers. Exa- T(3) , T(20), T(119), T(696), T(4059), T(23660),...... which are products of (2*3), (14*15), (84*85), (492*493), (2870*2871), (16730*16731), .... respectively. All such numbers have their last digit 0 or 6.If k,k+1 are consecutive numbers and R is the triangular number, then square root of (4R +1) in order to become an integer, Last digit of R should have either of 0,1,5,6. But since product of two consecutive numbers can have last digit either of (0,2,6), it implies R can have only 0 or 6 as last digit. Moreover, the consecutive numbers will have last digits (0,1) or ( (2,3) or (4,5). |
*A different sort of presence of π was
discovered by Jones Castillo Toloza in 2007 associated with triangular
numbers-- π - 2 = 1/1 +1/3 - 1/6 -1/10 + 1/15 + 1/21 -1/28 -1/36 + 1/45 + 1/55 -.......... or, π - 2 = 1/C(2,2) + 1/C(3,2) - 1/C(4,2) - 1/C(5,2) +1/C(6,2) -1/C(7,2) -1/C(8,2) - 1/C(9,2) +1/C(10,2) + 1/C(11,2) +.............. |
References -- Triangular Numbers |
1. www.shyamsundargupta.com |
2. www.cut-the-knot.org & www.facebook.com/alexander.bogomolny |