I noticed that the results for x increase then reset every 3 steps, 0-2, 3-5, 6-8, etc...except for 15-21, this pattern continues when n > 21. Just a simple observation, hopefully helpful, probably not though.
Here are the results for the values we know given those 2 mentioned formulas:
y = 2^n * x
and so x = y / 2^n
AND
y = 2^n + x
and so x = y - 2^n
| n | Known Results (y) | x = y / 2^n | x = y - 2^n |
| 0 | 1 | 1.00000000 | 0
|
| 1 | 3 | 1.50000000 | 1
|
| 2 | 7 | 1.75000000 | 3
|
| 3 | 8 | 1.00000000 | 0
|
| 4 | 21 | 1.31250000 | 5
|
| 5 | 49 | 1.53125000 | 17
|
| 6 | 76 | 1.18750000 | 12
|
| 7 | 224 | 1.75000000 | 96
|
| 8 | 467 | 1.82421875 | 211
|
| 9 | 514 | 1.00390625 | 2
|
| 10 | 1155 | 1.12792969 | 131
|
| 11 | 2683 | 1.31005859 | 635
|
| 12 | 5216 | 1.27343750 | 1120
|
| 13 | 10544 | 1.28710938 | 2352
|
| 14 | 26867 | 1.63983154 | 10483
|
| 15 | 51510 | 1.57196045 | 18742
|
| 16 | 95823 | 1.46214294 | 30287
|
| 17 | 198669 | 1.51572418 | 67597
|
| 18 | 357535 | 1.36388779 | 95391
|
| 19 | 863317 | 1.64664650 | 339029
|
| 20 | 1811764 | 1.72783279 | 763188
|
| 21 | 3007503 | 1.43408918 | 910351
|
| 22 | 5598802 | 1.33485842 | 1404498
|
| 23 | 14428676 | 1.72003222 | 6040068
|
| 24 | 33185509 | 1.97801048 | 16408293
|
| 25 | 54538862 | 1.62538475 | 20984430
|
| 26 | 111949941 | 1.66818412 | 44841077
|
| 27 | 227634408 | 1.69600850 | 93416680
|
| 28 | 400708894 | 1.49275696 | 132273438
|
| 29 | 1033162084 | 1.92441434 | 496291172
|
| 30 | 2102388551 | 1.95800192 | 1028646727
|
| 31 | 3093472814 | 1.44051053 | 945989166
|
| 32 | 7137437912 | 1.66181426 | 2842470616
|
| 33 | 14133072157 | 1.64530614 | 5543137565
|
| 34 | 20112871792 | 1.17072322 | 2933002608
|
| 35 | 42387769980 | 1.23364647 | 8028031612
|
| 36 | 100251560595 | 1.45885221 | 31532083859
|
| 37 | 146971536592 | 1.06935867 | 9532583120
|
| 38 | 323724968937 | 1.17770458 | 48847061993
|
| 39 | 1003651412950 | 1.82563129 | 453895599062
|