>>7485

>>7494

Thanks for these latest hints!

Pics attached for c145, c287, c288, and c6107 are an attempt to further explore the relationship between the product of (-f,1).a and (e,1).a and the corresponding factor record's aan(n-1) value.

Relevant columns are as follows:

• "a diff": difference between (e,1).a and (-f,1).a.

• "d diff": difference between (e,1).d and (-f,1).d.

• "aa product": (e,1).a * (-f,1).a

• "div 2": "aa product"/2. (abbreviated as aa/2 in this post)

• "ti": largest triangle base from aa/2 as calculated from the inverse triangle formula.

• "ti rm": remainder from aa/2 less the largest triangle.

• "square multiples": possible square factors of aa/2.

The aa/2 value is being analyzed because aan(n-1) = aa * 2T(n-1) for each factor record. Therefore we can isolate aa and T(n-1).

Each possible "square multiple" has it's own triangular breakdown. See the c6107 example for 4426800 that shows possible T formulas plus remainders for the 1, 2, 4, 5, 10, and 20 squares.

Might be going too deep here, but the objective is to see if there is a way to calculate the "aa product" for any factor record by understanding it's components.

Other observations on this data:

1) The "a diff" column matches the factor record's a value.

2) The "d diff" column can be used to navigate from the starting x value to the solution x:

If "d diff" is even, then starting x - "d diff" == solution x.

If "d diff" is odd, then starting x - "d diff" - 1 == solution x.

>>7571

There are a few limited cases where a factor record can be found directly from the triangle breakdown of the aa product.

Pics attached are for c119, c189, c275, c377, c495, and c629.

These numbers represent a small portion of integer sequence A181890 (oeis.org/A181890). (thanks SAGA for pointing this out)

In these cases, n-1 from the entry record is a perfect square, and the a value in the factor record is sqrt(n-1). Notice the matching ti and "div 2" columns.

>>7571

Have found a way to calculate a valid aa factor in (0,n) starting from the (-f,1) and (e,1) aa product.

Pics attached for c361, c961, c21025, and c144 show the previously explained aan(n-1) view, as well as the triangle breakdowns for each possible square multiple.

Recall that the aa product can be represented as aa * 2T(n-1) and potentially has multiple valid aa and T formula solutions.

For c361, as an example, the "aa product" is 26082 with possible square factors 1^2, 3^2, and 9^2. In the case of the 9^2 square multiple, the triangle formula can be represeted in terms of our starting x value of 18.

Triangle portion = T(x-1) + (x/2 - 1) = T(17) + 8 = 161

And solving for a possible aa factor:

26082 = aa * 2 * (T(17) + 8)

aa = 26082 / 322 = 81

a = 9

This similarly works for the other odd examples c961 and c21025, and finds that largest possible aa factor.

In the case of even c144, the methodolgy is the same, with a slight adjustment to the triangle formula:

Triangle portion = T(x-1) + (x/2 - 2)

So given the aa product from (-f,1) and (e,1), we are now able to determine the largest possible aa factor by calculating the triangle and it's remainder from x.