One year ago, I wrote:This of course raises the question: Is there a reccursion formula for N(R,C) for any R? How deep is it? Could this be used to determine PatmaxDaddy's 5xC-sequence further in just a fraction of a second?

1. Yes, most certainly

2. VERY deep

3. Definitely not, because (a) I don't think we will find the formula, and (b) still if we did, we would have to precalculate much more 5xC-values than we have now before the reccursion formula could be used.

All this because I'm sure I have found the 4xC reccursion formula. I have not proved it explicitely (and I don't think I will ever dear), but there are good reasons to believe it's correct.

Let R(C) be the number of RxC-bands divided by (4C)! * C!^12. From the experience with 3xC (the Franell numbers) it's reasonable to guess that R(C) fits into som reccursion forumla

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`F_0(C) * R(C) + F_1(C) * R(C-1) + ... + F_N(C) * R(C-N) = 0`

where the F_i are polynomials in C.

With some precalculated values of R(C), it's possible to experiment with systems of linear equations to get the polynomial coefficients.

This was best to do modulo a prime first, to get an idea of the reccursion depth and polynomial degrees. Afterwards, I did the calculation with integers and some more R(C) values than those just needed to get the coefficients. The linear system had one non-trivial solution for the guessed degree and depth. Some of the polynomials also had linear factors which fitted into the pattern I could sense in the Franell numbers formula.

For 3xC, depth was 2 and polynomial degree was 2. The coefficients were not bigger than 16.

For 4xC, depth seems to bee 13, polynomial degree is 8, and the biggest coefficient is 2906501472489100167249389420544.

No need to go for the 5xC formula...

The 4xC formula found is given in the next post.