Error in A115866

[N. J. A. Sloane]

sequence A090943  (even numbers n such that N(n) is squareful and 
  Any idea why these sequences are close to linear functions?
sequences 284+1000*n and 228+(10^4)*n; in the plot for a(n)<8000, the 
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Message-ID: <01f301c75dbb$b831fde0$a0933e44 at Dell>
From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
To: "Nick Hobson" <nickh at qbyte.org>
Cc: "seqfan" <seqfan at ext.jussieu.fr>
References: <8C8F8FDECD755A8-4F0-B12F at FWM-M05.sysops.aol.com> <8C8F9E2D56C2AB6-10B8-50F1 at FWM-M15.sysops.aol.com> <op.tldec2d2at030q at tata.domain_not_set.invalid> <op.tldsb7coui647s at eponymous> <op.tldu6hveat030q at tata.domain_not_set.invalid> <4597F95F.6040506 at indiana.edu> <op.tlhfm4kgat030q at tata.domain_not_set.invalid> <6.1.1.1.0.20070101212754.033bcb60 at mail.comcast.net> <000d01c72e82$6a722480$6401a8c0 at yourxhtr8hvc4p> <5542af940701021056u45862818j9976d1d6bbac9229 at mail.gmail.com> <op.tlk53rfoat030q at tata.domain_not_set.invalid> <459BDBDC.9070502 at wolfram.com> <op.tllcdaxaui647s at eponymous> <op.tnztbzj9ui647s at eponymous> <op.tolj10lzui647s at eponymous> <007601c75d68$9ee86200$a0933e44 at Dell> <op.tomhhruxui647s at eponymous>
Subject: Re: Error in A115866
Date: Sat, 3 Mar 2007 17:45:20 -0000
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On Saturday, March 03, 2007, "Nick Hobson" <nickh at qbyte.org> wrote:
> On Sat, 03 Mar 2007 07:50:30 -0000, David W. Cantrell
> <DWCantrell at sigmaxi.net> wrote:
>> On Saturday, March 03, 2007, "Nick Hobson" <nickh at qbyte.org> wrote:
>>> I believe there is an error in sequence A115866: "Number of paths
>>> from  (0,0,0) to (n,n,n) such that at each step (i) at least one
>>> coordinate  increases, (ii) no coordinate decreases, (iii) no
>>> coordinate increases by  more than 1, and (iv) all coordinates are
>>> integers. This is a  3-dimensional extension to A001850."
>>
>> For months, I had been planning to submit that very sequence!
>> Already
>> typed in a draft of an email to have been sent to Neil, my
>> (shorter)
>> title was to have been
>>
>
> I only found A115866 by accident.  Having found A001850, I initially
> hit  upon the erroneous extension to three dimensions mentioned
> below, and  hence A115866.  Then I realised that the boundary cases
> should be handled  as per the 2-D rather than the 1-D problem.
>
> Why not submit the 4-D version?!

Oh, I will. I had already written it up in the same draft email I
mentioned above, and had planned to submit it sequentially after the
3-D version, and followed sequentially by the 5-D version, etc.
stopping at... Well, I had intended to ask Neil where he thought I
should stop. So Neil, just tell me what dimension to stop at and I
will do so.

Among related submissions in that same draft email, there is a square
array in which the rows give the dimension N and the columns give n in
(n,n,...,n). Thus, in that square array, the second row is A001850,
the third row is A115866,

[P.S. Oops, I see now in a recent email from Neil that that A number
will not be correct. Apparently it will be A126086.]

etc. while the second column is the core sequence A000670, the third
column is A055203, etc.

I had even thought about submitting as a sequence the diagonal of that
square array

1, 3, 409, 10681263, 117029959485121, 939073157252309315848923,
8508048612432263410111274212273801489,...

under the name
Number of N-D paths from (0,0,...,0) to (N,N,...,N) using nonzero
steps of the form (x(1),x(2),...,x(N)) where x(k) is in {0,1} for 
1<=k<=N

David W. Cantrell




David,  strange things happen at about 8 or 9 dimensions,

of course the arrays and diagonal sequence(s) should
also be submitted!

Best

Neil