complementary sequences, triangular numbers, Fibonacci numbers

Kimberling, Clark ck6 at
Fri Mar 26 14:49:08 CET 2004

He SeqFans,


In Neil Sloane's "Classic Sequences" 

there are some "partial complements" whose union is the whole
complement.  For example, on page 1, the

Wythoff Array appears.  Look at column 3; its complement, C, is the
ordered union of columns 4,5,6,...


The sequence (2,3,5,8,13,...) in row 1 is the "1st partial complement"
of C;

the sequence (7,11,18,29,47,...) in row 2 is the "2nd partial
complement" of C; and so on.


In "Classic Sequences," scroll down to "Other Links" and visit "Partial
Complements and Transposable

Dispersions" (that takes you to JIS, where you can scroll down to
Article 04.1.6) - or,

for the pdf, just click )


Section 1 of the article defines "ith partial complement" for i =
1,2,3,...  The triangular numbers (and other

such sequences) are fixed points under certain mappings.  John and I
kicked this ball downfield aways 

but finally (p21) tossed up our hands and wrote "It is hoped that
someone will prove those conjectures."

(See the Abstract for a simple statement of the main conjecture.)


Clark Kimberling




-----Original Message-----
From: benoit [mailto:abcloitre at] 
Sent: Friday, March 26, 2004 12:03 AM
To: seqfan at
Subject:  Re: In One Sequence Or Another (Non-Fibonacci)


There is also the complement of triangular numbers with a simple formula
and some properties : 


Benoit Cloitre 

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