The Lost Sequence

I was playing around with sequences, and I thought I would screw around with the Shaw-Basho Polynomial:

I plugged 0 into the polyomial to get 4, and then I plugged in 1 to get 12, etc. I got the following infinite sequence of numbers:

4 12 35 89 213 511 1194 2622 5346 10150 18093 ... (goes on forever)

Not too interesting, eh? Then I wrote out the differences of succeeding numbers in the sequence. For example, 12 - 4 = 8, 35 - 12 = 23, 89 - 35 = 54, etc.

8 23 54 124 298 683 1428 2624 4804 7943 ... (goes on forever)

I kept doing this process. The third sequence began 23 - 8 = 15, 54 - 23 = 31, etc. When you keep going, something completely unexpected happens! Here - I've done the work for you:

SEQUENCE 1: 4 12 35 89 213 511
1194 2622 5346 10150 18093 ... (goes on forever)

SEQUENCE 2: 8 23 54 124 298 683 1428 2624 4804
7943 12458... (goes on forever)

SEQUENCE 3: 15 31 70 174 385 745 1296 2080 3139
4515 6250... (goes on forever)

SEQUENCE 4: 16 39 104 211 360 551 784 1059 1376
1735 ... (goes on forever)

SEQUENCE 5: 23 65 107 149 191 233 275 317 359
... (goes on forever)

SEQUENCE 6: 42 42 42 42 42 42 42 42 42 42...
(goes on forever)

SEQUENCE 7: 0 0 0 0 0 0 0 (goes on forever)

SEQUENCE 8: 0 0 0 0 0 0 0 (goes on forever)

SEQUENCE 9: 0 0 0 0 0 0 0 (goes on forever)

SEQUENCE 10: 0 0 0 0 0 0 0 (goes on forever)

And it stays at zero forever. The sequence destroys itself.

NOW: Look at the first element of each sequence, and you have the LOST numbers.
Weird, eh?

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