>>9245

Any triangle can be broken into 4 smaller and equal triangles, with a remainder, depending on u parity.

T(u) = 4*T(u') + rm

odd u

u' = (u-1)/2

rm = u - (u-1)/2

even u

u' = (u-2)/2

rm = u + u/2

Pic attached for u10025 - 10050 shows these results.

reviewing polite numbers again.

> Sum of polite numbers is the same as their difference of squares.

The count of numbers in a polite range is either one of the factors of c or twice that factor.

The sum of the first and last numbers in a polite range is also either one of the factors of c or twice that factor.

Examples attached for c34117=109x313, c260891=317x823, and c208255331=5449x38219.

The "(first+last)/2" column only divides even numbers by 2.

>>9294

and c208255331.

Posting some previous work which explored finding non-trivial records in the grid for multiplication.

Pic attached shows the results for c*d multiples of records between c300 and c400.

The trivial record is in the "c=cd" column.

The non-trivial record is in the "a,b" column.

Non-trivial records are created using the "formula" column. The rules can be summarized as follows:

odd c * odd d =a=d, b=c

odd c * even d, d%4=0 =a=d/2, b=2c

odd c * even d, d%4=2 =a=d/2, b=c

even c * even d =a=d, b=c

even c * odd d =a=2d, b=c/2

The d%4=2 exception is because the resulting c will also be mod 4 = 2, and must be divided by 2 to create a valid record.

>>9325

Given e and t in row 1, we can calculate d, f, and j.

even e, d = 4T(t-1) + e/2

odd e, d = 2(t^2) + (e-1)/2

f = 2d+1-e.

j = sqrt(f)

Subsequent e columns can be found by looping until j=1:

j = j - 1;

d = d - 1;

f = j * j;

e = 2 * d + 1 - f;

First pic attached shows the list of records starting from (1,1,5) where a=41. The last 2 records correspond to (2a-1,1,1) and (2a,1,1).

Also attached are examples of e columns with matching a values in (1,1) and (2,1) for t=1 to 10.