[seqfan] Re: More sequences

[Richard Guy]

I apologize for sending two messages on the same day, but
I think that many can learn from my reactions to the
reactions to my previous message.  These consisted (so far)
of one comparatively kneejerk and trivial comment, and
a telephone call (!) from the Great One himself, who said
that I ought to submit sequences myself.  I illustrate
below the difficulties with the latter.  I don't write
programs, so that any submission that I might make is much
shorter than need be, yet also likely to be error-ridden.
It's a pity that most of us are interested in our own pet
sequences, but rarely in those of others.  Many of you
could produce hundreds of terms with little effort and
no mistakes.

So I go on calculating my own current pet.  I don't know how
to write code, so it's all done by hand, and, as far as I've
gone, almost certainly contains at least one error, so that
I'm wasting my time and that of anyone else who bothers with it.
But one learns something:--

1, 3, 6, 10, 15, 21,  (differences  2, 3, 4, 5, 6  so far, but
then)  4, 5, 11, 14, 2, 7, 9, 16, 20, 29, 35, 46, 18, 31, 33,
48, 52, 12, 13, 23, 26, 38, 43, 57, 24, 25, 39, 42, 22, 27, 37,
44, 56, 8 (at last), 17, 19, 30, 34 (now we have everything
but 28, 32, 36, ...), 47, 53, 28 (and now only consecutively
increasing squares are available), 36, 45, 55, 66, 78, 91, 105
(giving another bunch of triangular numbers with differences
8, 9, 10, 11, 12, 13, 14), 64, 80, 41, 40, 60, 61, 83, 86, 58,
63, 81, 88, 108, 117, 79, 65, 104, 92, 77, 67, 54, 90, 106,
119, 50, 71, 73, 96, 100, 69, 75, 94, 102, 123, 133, 156, 168,
121, 135, 154, 170, 191, 98, 127, 129, 160, 164, 32 (at last!
and now we recruit several holdouts:), 49, 51, 70, 74, 95,
101, 68, 76, ...

so that the answers to my questions:  does every number appear?
and:  do we ever again get the complete set of the first  n
numbers?  are almost certainly  yes!  and  no!  respectively,
but can you prove them ?     Best wishes,   R.