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Thursday, December 31, 2009

A Sequence of Sequences

This research was supported in part by a grant from the Universal Walloons and by a grant from the Ants of America. We thank them for their help.

Neither this infinite sequence of sequences nor any of the component sequences (which are all finite) is an unfolding sequence, but you will notice reminiscent properties. How is this generated? For component S(n), consider all words composed of "1" and "2" of length 0-n. Left justify. Sort. S(n) is the lengths of the words in sorted order. Thus S(2): Possible words are -,1,2,11,12,21,22. In sorted order, they are -,1,11,12,2,21,22. Their lengths comprising S(2) are 0,1,2,2,1,2,2.


0....................................................................0- 0

011..................................................................0- 2

0122122..............................................................0- 6

012332331233233......................................................0- 14

0123443442344344123443442344344......................................0- 30

012345545534554552345545534554551234554553455455234554553455455......0- 62

0123456656645665663456656645665662345665664566566345665664566566.....0- 63
123456656645665663456656645665662345665664566566345665664566566.....64-126

0123456776775677677456776775677677345677677567767745677677567767.....0- 63
7234567767756776774567767756776773456776775677677456776775677677....64-127
1234567767756776774567767756776773456776775677677456776775677677...128-191
234567767756776774567767756776773456776775677677456776775677677....192-254

0123456788788678878856788788678878845678878867887885678878867887.....0- 63
8834567887886788788567887886788788456788788678878856788788678878....64-127
8234567887886788788567887886788788456788788678878856788788678878...128-191
8345678878867887885678878867887884567887886788788567887886788788...192-255
1234567887886788788567887886788788456788788678878856788788678878...256-319
8345678878867887885678878867887884567887886788788567887886788788...320-383
2345678878867887885678878867887884567887886788788567887886788788...384-447
345678878867887885678878867887884567887886788788567887886788788....448-510

etc.

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