>>3288
>>3282
Thanks anon, this was an interesting dive, only part way through understanding (haven't dug into patterns at n=1).
>>3290
>But how is this a solution when the a and b values are still unknown?
>What am I missing?
Don't think you're missing anything, and not sure yet either, but useful for a quick jump to n=0 or n=1.
It also produces a new e that is a perfect square. The new t is always an integer that is the square root of the negative of the new e. This is expected because we know:
e=2an-x^2, and if n=0, e=-x^2
I think being able to transform to a perfect square column in the -e space may be useful once everything is worked out. There is a subsequent transform of 'a' needed, as illustrated below.
Attached images relevant.
First is the orig VQC, generated by Teach back in Dec and on pastebin. Showing only the -e perfect square columns, for n=0 and n=1. The gray bar is the first record, passing through {0:0:1:0:1:1}. Added x's starting at e=-9 above bar, as a sequence for 'a' could be extended upward with decreasing a values (moving t through zero to negative?).
Note: at n=0: d, a, and b increment by 1 for each t. x never changes. x^2=-e. There is a gap between a and b, which is 2*x, and this never changes for t sequences. Also, d-x=a for any n=0.
Note: observe how d increases as we move leftward decreasing e (each +1 increase in x). It's not linear.
seq:1,2,3,5,9,13,19,25,33
See: https:/ /oeis.org/A085913
A085913: Group the natural numbers such that the product of the terms of the n-th group is divisible by n!. (1),(2),(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),… Sequence contains the first term of every group.
1, 2, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405
There is a similar (same initially) sequence
See: https:/ /oeis.org/A099392
A099392: a(n) = floor((n^2-2*n+3)/2).
1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405
The second formula was easier in excel, but it breaks down at x=54. The d, a, b, and c get out of sync, and no longer check out (see highlighted region in screen cap). Maybe that's why the oeis.org sequence ends there? Maybe the A085913 sequence based on the factorial and groups would continue to work? Not sure, but this doesn't scale at this point for d, using this quick excel model.
- also highlighted in that image is e=-1849, the transform for 87 (1x87) to n=0: {-1849:0:44:43:1:87}, if anyone wants to find a link to the 87 (3x29) record transformed to n=0: {-169:0:16:13:3:29}
Ok, here are transformation examples with a few of the 'known' factors we've been using. Doesn't apply to getting from the 1*c to desired a*b factors yet, but interesting patterns. Refer to image with A085913 in top left, starting at e=0.
Take 85 (5x17)
{4:2:9:4:5:17}
transformed to n=0 is:
{-36:0:11:6:5:17}
Calc d for t=1 at x=6 is 19 (highlighted tan), and we want d=11, so transform 8 steps. a and b are 13 and 25, so taking away 8 from each is 5 and 17, 5x17=85, check.
87 (3x29)
{6:7:9:6:3:29}
to
{-169:0:16:13:3:29}
Calc d for t=1 at x=13 is 85 (highlighted tan), and we want d=16, so transform 69 steps. a and b are 72 and 98, so taking away 69 from each is 3 and 29, 3x29=87, check.
145 (5x29)
{1:5:12:7:5:29}
to
{-144:0:17:12:5:29}
Calc d for t=1 at d=12 is 73 (highlighted tan), and we want d=17, so transform 56 steps. a and b are 61 and 85, so taking away 56 from each is 5 and 29, 5x29=145, check.