PELL Numbers Formulae
You have reached the PELL Numbers Formulae Page of RAJESH RAM (maths@nivedh.com). Click here for a GRAPHICS Version of this page. Click here for my mathematics home page
Other Pages : PELL Numbers Formulae , Sums of Powers, FIBONACCI Numbers Formulae, TRIANGLE Numbers that are Perfect Squares, Sums of Cubes, Sums of Powers - Ramanujan and his Number 1729A brief introduction to PELL Numbers : 1,2,5,12,29,70,169,408,,....... which is given by
P(k) = 2P(k-1) + P(k-2), P(0)=0, P(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.
| P(k) = SIGMA(m=1 to [(k+1)/2]) 2^(k-2m+1) * NCR(k-m, m-1) | P1 |
| SIGMA P(k) = SIGMA(m=1 to k) 2^(m-1) * SIGMA(j=0 to [(k-m)/2]) NCR(m-1+j, j) | P5 |
| SIGMA (-1)^k * P(k) = SIGMA(m=1 to k) (-2)^(m-1) * SIGMA(j=0 to [(k-m)/2]) NCR(m-1+j, j) | P8 |
| P(k) ^ 2 = SIGMA(m=1 to k) 4^(k-m) * SIGMA(j=1 to m) (-1)^(j-1) * NCR(2k-m-j, m- j) | P9 |
Note : See "TRIANGLE NUMBERS THAT ARE PERFECT SQUARES" to see its relation with PELL Numbers and how you can use that formulae to get more formulae for PELL Numbers