| m= | Floor(Sqrt(m)) | Period of Continued Fraction for Sqrt(m) |
|---|---|---|
| n2+3n+2 | [n+1] | [2,2n+2] |
| 25n2+14n+2 | [5n+1] | [2,2,10n+2] |
| 36n2+17n+2 | [6n+1] | [2,2,2,12n+2] |
| 841n2+82n+2 | [29n+1] | [2,2,2,2,58n+2] |
| 1225n2+99n+2 | [35n+1] | [2,2,2,2,2,70n+2] |
Example: Row 2 with n=2: the continued fraction expansion for the square root of 130 is [11,2,2,22,2,2,22,2,2,22,...].
Unlike the "all 1's" case, we can find an infinite family of integers m with the palindromic part of the period in the continued fraction expansion of the square root of m consisting of ANY number, k, of 2's. We prefer, however, to write two formulas for m: one if k is even:
| m = Pk+12 n2 + 2 ( Pk+1 + Pk) n + 2 |
and one when k is odd:
| m = (Pk+12/4) n2 + ( Pk+1 + Pk) n + 2 |
where Pk is the k-th Pell number. See also Sloane's A000129.