Pell Number - Computations and Connections

Computations and Connections

The following table gives the first few powers of the silver ratio and its conjugate .

0
1
2
3
4
5
6
7
8
9
10
11
12

The coefficients are the Half companion Pell numbers and The Pell numbers which are the (non-negative) solutions to . A Square triangular number is a number which is both the th triangular number and the th square number. A near isosceles Pythagorean triple is an integer solution to where .

The next table shows that splitting the odd number into nearly equal halves gives a square triangular number when n is even and a near isosceles Pythagorean triple when n is odd. All solutions arise in this manner.

t t+1 s a b c
0 1 0 0 0 0
1 1 1 0 1 1
2 3 2 1 2 1
3 7 5 3 4 5
4 17 12 8 9 6
5 41 29 20 21 29
6 99 70 49 50 35
7 239 169 119 120 169
8 577 408 288 289 204
9 1393 985 696 697 985
10 3363 2378 1681 1682 1189
11 8119 5741 4059 4060 5741
12 19601 13860 9800 9801 6930

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