Hofstadter's Triangles between Squares

[Michael Somos]

                                                          19 Apr 2000

This is a little excerpt of a recent book by Hofstadter and others
titled "Fluid concepts and creative analogies". The first chapter
is "To Seek Whence Cometh a Sequence". He describes a program named
"Mathgod". He also mentions on page 25 that at age sixteen :

-------------------------------------------------------------------------
    I soon wrote a computer program -- my first serious program ever --
  that verified the hypothesized identicality of the derived sequence
  and the original sequence for several thousand terms. At that point,
  no doubt was left in my mind that I had unlocked the secret of the
  distribution of the triangular numbers among the squares. (A few
  months later, I also proved this result, after slowly working out
  the appropriate concepts with which to do it.)
    This discovery launched a several-year period in my life during
  which I becamse truly obsessed with integers sequences. Over these
  years I invented hundreds of sequences, many of them having complexly
  tangled recursive properties that made this first one look almost
  trivial. And the variety of mathematical ideas that went into the
  sequences was also huge. But there was something beautiful about
  this "first love" that no other discovery ever quite equaled.
-------------------------------------------------------------------------

I think he deserves to be placed in a "Sequences Hall of Fame" for
this description. By the way, the sequence he mentions is A005214
and it first appears on page 15 as the following :

-------------------------------------------------------------------------
         1, 1, 3, 4, 6, 9, 10, 15, 16, ...

    Note that 1, being both a square and a triangular number,
    is included twice, wearing its "square" hat to the left of its
    "triangular" hat.
-------------------------------------------------------------------------

Compare this with the version in EIS :

-------------------------------------------------------------------------
%I A005214
%S A005214 1,3,4,6,9,10,15,16,21,25,28,36,45,49,55,64,66,78,81,91,100,105,120,
           121,
%T A005214 136,144,153,169,171,190,196,210,225,231,253,256,276,289,300,324,325,
           351,
%U A005214 361,378,400,406,435,441,465,484,496,528,529,561,576,595,625,630,666,
           676 
%N A005214 Triangular numbers together with squares.
%D A005214 Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer
           Models of the Fundamental Mechanisms of Thought".
%K A005214 nonn
%O A005214 1,2
%A A005214 Russ Cox (rsc at research.att.com)
-------------------------------------------------------------------------

I suggest including the missing "1" since Hofstadter does have it.

-------------------------------------------------------------------------
%S A005214 1,1,3,4,6,9,10,15,16,21,25,28,36,45,49,55,64,66,78,81,91,100,105,120,
           121,
%D A005214 Hofstadter, D. R., Fluid Concepts and Creative Analogies: Computer
           Models of the Fundamental Mechanisms of Thought, (together with the
           Fluid Analogies Research Group), NY: Basic Books, 1995. p. 15.
%O A005214 1,3
-------------------------------------------------------------------------

Shalom, Michael

-- 
Michael Somos <somos at grail.cba.csuohio.edu>     Cleveland State University
http://grail.cba.csuohio.edu/~somos/            Cleveland, Ohio, USA 44115