# [seqfan] An interesting coincidence

Jeremy Gardiner jeremy.gardiner at btinternet.com
Sun Dec 28 18:50:04 CET 2008

```It strikes me as an interesting coincidence that the following sequences
appear to have the same underlying parity.

There may be a trivial explanation, however, "The existence of a coincidence
is strong evidence for the existence of a covering theory"!

"Are There Coincidences in Mathematics?"
Philip Davis
The American Mathematical Monthly, vol. 88, 1981, pp. 311-320.

http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2945

Jeremy Gardiner

A003071    Maximal number of comparisons for sorting n elements by list
merging.
A029886    Convolution of Thue-Morse sequence A001285 with itself.
A061297    a(n) = Sum_{ r = 0 to n} L(n,r), where L(n,r) (A067049) = LCM (n,
n-1, n-2, ..., n-r+1)/ LCM ( 1,2,3,...r).
A092524    Binary representation of n interpreted in base p, where p is the
smallest prime factor of n: p=A020639(n).
A093431    a(n)=sum_{k=1..n}(LCM(n,n-1,...,n-k+2,n-k+1)/LCM(1,2,...,k)).
A102393    A wicked evil sequence.
A104258    Replace 2^i with n^i in binary representation of n.
A122248    a(n)-a(n-1)=A113474(n).
A128975    a(n) = the number of unordered triples of integers (a,b,c) with
a+b+c=n, whose bitwise XOR is zero.
Equivalently, the number of three-heap nim games with n stones
which are in a losing position for the first player.

A003071
,0,1,1,1,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,0,1,0,1
,1,1,0,1,1,1,1,1,0,1,0,1,1,1,1,1,0,1,1,1,0,1,0,
A122248
,0,1,1,1,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,0,1,0,1
,1,1,0,1,1,1,1,1,0,1,0,1,1,
A093431
,1,1,1,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,
A128975
,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0
,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0
,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,
A102393
,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0
,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0
,0,0,1,0,1,0,0,0,0,0,
A029886
,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0
,0,0,1,0,0,0,0,0,1,0,1,0,0,0,
A092524
,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0
,0,0,1,0,0,0,0,0,1,0,
A061297
,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0
,0,0,
A104258
,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,

```