User:Danny Rorabaugh

Danny Rorabaugh is a postdoctoral fellow at Queen's University Sep 2015 -- Dec 2017, the University of Delaware Jan -- May 2018, and the University of Tennessee Knoxville Jun 2018 -- present.

Words

My dissertation at the University of South Carolina was Toward the Combinatorial Limit Theory of Free Words.

The sequences most relevant to that words research involve...

... Zimin words: A082215 (see Kamenetsky's formula) and A123121 (but actually A000225);

... words with(out) a palindrome/square/bifix: A003000(A094536), A019308(A094538), A248122(A252696), A249629(A252697), and A249638(A252698) thru A249643(A252703);

... word structures: A000110, A056272-A056273, A164863-A164864, A203641, A255706 (my favorite submission).

Puzzles

Here are a few puzzle sequences I've created that are not in the OEIS (at the time of posting). Please message me if you've figured out a pattern.

[2015-04] 1,1,1,1,2,1,2,1,5,3,2,2,2,1,2,2,1,1,5,2,1,2,1,1,15,...

[2017-09] 1,2,3,5,11,37,277,4907,...

[2017-09] 5,4,3,4,7,5,3,4,4,5,4,6,6,9,11,9,...

[2017-09] 2,3,5,7,9,11,15,16,17,19,21,22,...

[2017-09] 1,4,6,10,12,18,22,28,32,42,46,58,...

Tables

I don't care much for sequences that are tables, but I do like a good table of sequences.

Cecilia Rossiter posted several sequences (most of A101089-A101104) involving the function MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n, for which I made the following table. Please email me if you find any inaccurate cross-references (other than shifted offset) or sequences that fill the holes.

Note: filling in holes in a table like this is NOT sufficient reason to submit a new sequence!

...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8 
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | ....... | ....... 
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | ....... 
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | ....... 
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | ....... 
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182 
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178 
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524 
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016 
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542 
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636 
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642 
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647 
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......