>>6654

>When does c first appear at a[t]?

>When is the second time it appears?

Not sure about the first time yet, but the second time is always in the same (e,n) at t = t + n.

Stumbled upon the third time c appears at a[t].

Borrowing from QVC's variable style >>6736, the following formulas represent the move to the third occurrence for odd e only, starting from the 1,c entry record:

e' = e

n' = 2x + n + 2

x' = 3x + 2n + 2

a' = b

d' = a' + x'

b' = a' + 2x' + 2n'

t' = (c+1)/2 + t

Examples:

entry

(1,61,6) = {1:61:12:11:1:145} = 145

2nd a[t]=c

(1,61,67) = {1:61:278:133:145:533}

3rd a[t]=c

(1,85,79) = {1:85:302:157:145:629}

n' = 2*11 + 61 + 2 = 85

entry

(31,128,8) = {31:128:16:15:1:287} = 287

2nd a[t]=c

(31,128,136) = {31:128:558:271:287:1085} = 311395 (second occurrence)

3rd a[t]=c

(31,160,152) = {31:160:590:303:287:1213} = 348131 (third occurrence)

n' = 2*15 + 128 + 2 = 160

Not sure if this is useful other than to show an example of how n can be calculated in terms of x and n.

>>6881

Figured out the indexing problem for even e to find the 3rd occurrence of a[t] = c, starting from the original c entry record.

Formula for n stays the same:

n' = n + 2x + 2 .

t shortcuts are as follows:

odd e: t' = n' - t

even e: t' = n' + 1 - t

The attached pic shows the starting c record and the 2nd and 3rd occurrences for a number of test cases.

The remaining columns show the u values from the 2nd a[t] and 3rd a[t], their difference, and a number of ways to represent that difference.

u' = u + 2d'

u' = u + 2(x'+1)

or, without going back to the starting record at all:

u' = u + 2*(sqrt(a''))

Where ' is the 3rd record, is the 2nd record, and ' is the starting record. (Please excuse the notation).

>>6619

>>6643

>>6858

Continuing with the gcd approach from the previous thread, and based a little bit on VQC's function idea, pics attached represent a comparison matrix between a[t] and a[p+1-t] records in both (-f,1) and (e,1).

Examples attached are for c145, c297, c6107.

Starting from easy to calculate records (starting c, na, nb, nc, 2nd a[t]=c, 3rd a[t]=c) the comparison shows for both -f and e columns:

1) The record at t=p+1-t based on each factor (p).

2) the gcd(a[t], a[p+1-t]) result.

3) the different between t values.

4) the gcd(c, t diff) - noticed that sometimes this gives the correct factor.

Records are displayed just showing (e,n,t) index, and the relevant a value used in the calculations (just to save space).

The gcd(-f,e) column at the end is the result of gcd( (-f, 1) a[t], (e,1) a[t] ), and records are filtered to where any of the gcd results matches p.

Given that these comparisons are against offsets of known factor values (p), there are a couple of things that stand out:

a) The gcd solution sometimes comes from the (-f,1) column and sometimes from (e,1).

b) For c6107, the factor shows up when comparing (e,n) a[t] instead of (e,1) a[t].

c) For c145, funny enough, the factor shows up in the very first gcd calculation between the na record an it's -x offset.

Think the point of this post is to perhaps illustrate that we have almost enough records to solve the problem.

>>6929

Found and fixed a bug.

The negative x offset formula for a[p+1-t] needs to be adjusted depending on odd or even e.

odd e: a[p+1-t]

even e: a[p+2-t]

Based on this change, attached are revised "matrix" views for c145, c287, c6107 and a new test for c96133.

The results are now more consistent across both -f and e columns with gcd solutions being found for various a[t]=c records.

>>6943

>Then we will look at the key that is made by column -f with the locations of c in (-f,1)

Might be able to add a bit of clarity here.

Attached are the a[t]=c relevant records for c145 in both e and -f, and then the na transforms for each in the c145-occurrences-na image.

We can currently get to all of these records reliably except the first occurrence of a[t] = c, labeled as "1st_c_at".

For c145, those 1st occurrence records are (1,1,9), (-24,6,22), and their corresponding na records of (1,1,9) and (-24,1,22).

Notice that the other records between e and -f columns have the same t values +/- 1.

What causes the first occurrence records to behave differently in -f vs e?

How do we determine the first occurrence?

Don't have these answers yet, but do believe this is the key VQC was talking about.

>>6654

>When is the first time a squared appears?

>What is the factor it is multiplied by?

>What is the first time b squared appears?

>The second?

Circled back to VQC's Independence Day post regarding the first and second occurrences of a^2 and b^2 in e and -f.

Attached pics are test cases for c145, c287, and c551 and show the first 2 records where a[t] mod a^2 or b^2 = 0, and a[t] is multiplied by another factor, in both positive and negative x.

The output shows -f details on the left and e on the right, and includes:

p column is the [p+t] formula where p is either a^2 or b^2

px column is the [p+1-t] formula, same a^2 or b^2 values.

n=a[t]/p just shows how n is derived for a[t] for known factors.

The p and px columns again show valid jumps between sequences of records.

The reason for this post, however, is that there is a pattern between the two a^2 and two b^2 records that enables us to predict the correct t values.

For example, in c145:

between (-24,1,10) and (-24,1,17)

10 + 17 - 2 = 25

between (-24,1,312) and (-24,1,531)

531 + 312 - 2 = 841

between (1,1,22) and (1,1,29)

((22 + 29) - 1)/2 = 25

between (1,1,821) and (1,1,862)

((821 + 862) - 1)/2 = 841

And the generalized formulas:

even e:

t1 + t2 - 2 = p

odd e:

(t1 + t2 - 1)/2 = p

In the case of odd e especially, it's interesting to note that the genalized t1 and t2 formula gives different results than the p and px pointer formulas.

>>6980

>>6983

Thanks for double checking.

In reviewing further, I think I may have just restated how the p+t formulas work in a different way.

Was looking for a way to jump sequences, but don't believe these formulas enable that kind of movement.

Not quite sure where to go with the sqrt(2d) hints, so in the meantime have integrated the iterative search solution for both odd/even squares into (-f,1) and (e,1) records.

Attached pics are test cases for c85, c145, and c15120 and show each factor record next to the (-f,1) or (e,1) record with a matching u.

For even e:

• the XPN is calculated as 1+8T(u).

• the corresponding n=1 record can be found at t = u + 1.

For odd e:

• the XPN is calculated as 4T(u) + 4T(u-1).

• the corresponding n=1 record can be found at t = u.

u is simply iterated from 0 to a max value from the entry record.

Rather than being a pathway to a solution, this was just an exercise to integrate the even square triangle formulas and visualize perhaps how factor records "bounce" between -f and e (c15120).