Fascinating Palindrome Connection (2)

The approach in the last entry can be generalised for all self-generating numbers, where in the appropriate number bases, the original starting number * k (where k = 2, 3, 4, 5, …) = its reverse number.

When n = 4, a unique repeating 2-digit sequence applies to 1/n in bases 2n – 1, 3n – 1, 4n – 1, …, so
that the original starting number (using these 2 digits) * 3 = its reverse number.

And just as the k^th term in the previous case “Fascinating Palindrome Connection 1”, where n = 3, is given by 3n² – 2n, the k^th term (where n = 4) is given by 4n² – 2n, resulting in the series,

2, 12, 30, 56, 90, …

So the relevant number bases, where this form of self-generating behaviour applies is in the number bases 7, 11, 15, … (Once again we omit the number base where n – 1 = 3, applying to the 1^st term of the series i.e. 2, as this results in a redundant 1^st digit of 0 in the original starting number).

However the 2^nd term, applying to base 7 is fully valid.

And 12 in base 7 = 15.

Therefore, 15 * 3 = 51 (in base 7).

However this can equally be expressed as 2 * (2n – 2) = 2 * 6 = 12 (in base 10), i.e. 15 in base 7.

So the subsequent 2-digit starting terms are given by 3 * 10 (= 28 in base 11), 4 * 14 (= 3B in base 15), 5 * 18 (= 4E in base 19) and so on.

Thus  28 * 3 = 82 (in base 11)

         3B * 3 = B3 (in base 15)

         4E * 3 = E4 (in base 19)

                  …

Then one 3-digit term in each case is obtained from 22 * 6 in base 7, 33 * 10 (in base 11), 44 * 14 (in base 15), 55 * 18 (in base 19) and so on.

And as we saw in “Fascinating Palindrome Connection 1” one can then use the formulas k/2(k/2 + 1), where k is odd and k/2 * k/2 where k is even, to calculate the total number of self-generating numbers (of a particular type) up to (and including) k digits.

So for example the total collection of self-generating numbers (where the original starting number * 3 = its reverse) -up to an including 4 digits - in each relevant number base, is    

4/2 * 4/2 = 4.

And in base 7, these are 15, 165, 1515 and 1665 respectively.