A006072 and A111065

[JEREMY GARDINER]

As commented in the OEIS, A006072 and A111065 appear to have the same parity (checked up to n=80).

 

Sequence A006072(n):
0,1,8,11,88,101,111,181,808,818,888,1001,1111,1881,8008,8118,8888,10001,10101,10801,11011,11111,11811,18081,18181,18881,80008,80108,80808,81018,81118,81818,88088,88188,88888,100001,101101
Name: Mirror symmetry about middle.

 

Sequence A111065(n):

0,69,96,609,906,6009,6699,6969,9006,9696,9966,60009,66099,69069,90006,96096,99066,600009,606909,609609,660099,666999,669699,690069,696969,699669,900006,906906,909606,960096,966996,969696,990066,996966,999666
Name:      Numbers that look the same when printed upside down (Numbers with 1's and 8's are excluded).

 

Parity:

0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1, ...

 

Q1. why should this be the case?

It's not obvious to me, however the fact that both sequences are related to particular symmetries makes me think it's more than a coincidence.

 

This may be a clue: The generating sequence for the number of 0's and 1's of the parity sequence is:

 

1,1,1,1,1,3,3,3,3,9,9,9,9,27, - This sequence is not in the OEIS.

 

Q2. Is the sequence A(n) = Powers of 3 repeated 4 times, "interesting" enough to be submitted to the OEIS (given the parity relationships above)?

 

Jeremy Gardiner

 
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