TRIANGLE Numbers that are Perfect Squares
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Other Links : PELL NUMBERS FORMULAE , SUMS OF POWERS, FIBONACCI FORMULAE, RAJESH's MATHS PAGE, TRIANGLE NUMBERS THAT ARE PERFECT SQUARES
Also see Sums of Cubes , Sums of Powers - Ramanujan and his Number 1729
T^2 = Y(Y+1)/2 = SIGMA Y
The solutions to the above equation is given by Sets T and Y where
T = {1, 6, 35, 204, 1189, 6930, .........}
Y = {1, 8, 49, 288, 1681, 9800, ........}
So, we can express the solution using the following relations:
T(k) = 6T(k-1) - T(k-2), T(0)=0, T(1) =1
Y(k) = 6Y(k-1) - Y(k-2) + 2, Y(0)=0, Y(1) =1
The following are the notations used :
1) SIGMA(m=1 to k) = Sum from m=1 to m=k
2) NCR(n,r) = n Choose r = n! / (r! (n-r)!)
3) [n] = integer part of n, for example [2.5] = 2
4) a^b = a to the power of b
5) a*b = a multiplied by b. Note that I have used * only to separate terms and not for
every case like 2k = 2*k
6) |a| = absolute value of a, for example |-1| = 1
The following are some of my findings that give formulae to find the kth, 2kth, etc.
| T(k) = SIGMA(m=1 to k) 4^(k-m) * NCR(2k-m, m-1) | T1 |
| T(k) = SIGMA(m=1 to [(k+1)/2]) (-1)^(m-1)*6^(k-2m+1)*NCR(k-m, m-1) | T2 |
| T(k) = SIGMA(m=1 to k) (-1)^(m-1)* 8^(k-m)*NCR(2k-m, m-1) | T3 |
| SIGMA T(k) = SIGMA(m=1 to k) 4^(k-m)*NCR(2k+1-m, m-1) | T4 |
| SIGMA T(k) = SIGMA(m=1 to k) (-1)^(m-1)* 8^(k-m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k+1-m-j, m-j) | T5 |
| SIGMA T(k) = SIGMA(m=1 to k) 6^(k-m) * SIGMA(j=1 to [(m+1)/2]) (-1)^(j-1)*NCR(k-m+j-1, j-1) | T6 |
| T(k)^2 = SIGMA(m=1 to 2k) 4^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j, m-j) | T7 |
| T(k)^2 = SIGMA(m=1 to 2k) (-1)^(m-1)* 8^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j, m-j) | T8 |
| T(k)^2 = SIGMA(m=1 to k) (-1)^(m-1)* 36^(k-m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k-m-j, m-j) | T9 |
| T(2k) = 6 * SIGMA(m=1 to k) 32^(k-m) * NCR(2k-m, m-1) | T10 |
| T(2k) = 6 * SIGMA(m=1 to [(k+1)/2]) (-1)^(m-1) * 34^(k-2m+1) * NCR(k-m, m-1) | T11 |
| T(2k) = 6 * SIGMA(m=1 to k) (-1)^(m-1) * 36^(k-m) * NCR(2k-m, m-1) | T12 |
| (T(k) + T(k-1))^2 = SIGMA(m=1 to 2k-1) (-1)^(m-1) * 8^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j-2, m-j) | T13 |
| (T(k) - T(k-1))^2 = SIGMA(m=1 to 2k-1) 4^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j-2, m-j) | T14 |
| Y(k) = SIGMA(m=1 to k) (-1)^(m-1)* 8^(k-m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k-m-j, m-j) | T15 |
Note : T is a actually contained in Pell Numbers P = {1, 2, 5, 12, 29, 70, 169, 408, ......} in more than one way. If you take the even terms out of P and write it as {2, 12, 70, 408, .....} = 2 {1, 6, 35, 204, .....} = 2T. This means that you can substitute (1/2)P(2k) in all the formulae for T(k) to get PELL Numbers Formulae. Also consider Pell numbers P = {1, 2, 5, 12, 29, 70, .....} and its associated numbers Q = {1, 3, 7, 17, 41, .....} The Pell numbers P(n) and their associated numbers Q(n) satisfy P(n+2)=2P(n+1)+P(n), P(0)=0, P(1)=1; Q(n+2)=2Q(n+1)+Q(n), Q(0)=1, Q(1)=1. Note that P(n).Q(n) = T(n). So this means that we can arrive at formulae for Q(n) since we have formulae for T(n) and P(n).