[seqfan] Re: editing A000217 Triangular numbers (long message)

hv at crypt.org hv at crypt.org
Fri Mar 12 11:54:58 CET 2010

Joerg Arndt <arndt at jjj.de> wrote:
:My questions and remarks regarding A000217,
:your comments are most welcome:

Gosh, what a mess. Thanks for trying to bring it under control.

: Number of ways a chain of n non-identical links can be be broken
: up. This is based on a similar problem in the field of proteomics:
: the number of ways a peptide of n amino acid residues can be be
: broken up in a mass spectrometer. In general each amino acid has a
: different mass, so AB and BC would have different masses. - James
: Raymond (raymond(AT)unlv.edu), Apr 08 2003
:- I fail to understand (delete?, but see example below)

>From the example below, it's clear it intends the number of different
fragments you can obtain from breaking up a chain of n distinguishable
items, i.e. n length-1 fragments + (n-1) length-2 fragments + ... + 1
length-n fragment.

I am neutral on whether it should be kept. If so, however, then it needs
rewriting, and as you suggest the example must move to be a part of the
comment. (To be fair, I guess the separation of comment and example was
less obviously a problem when there were just a handful of commments.)

: Number of distinct straight lines that can pass through n points in
: 3-dimensional space. - Cino Hilliard (hillcino368(AT)gmail.com), Aug
: 12 2003
:- I fail to understand (delete?)

Presumably, the C(n, 2) ways you can pick two of n points to draw
a line between them. Assuming either that each line is a segment
beginning and ending at one of the points, or that no 3 points are
colinear. Looks like a delete to me.

: The number of distinct handshakes in a room with n people (n>=2). -
: Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 12 2007
:- wrong, n+1 people.  I'll correct.

This seems dubious to me - it needs a lot more clarification of a
"handshake" to make sense, all of which makes it just a clearer
obfuscation of C(n, 2).

: a(n), n>=1, is the number of ways in which n-1 can be written as a
: sum of three positive integers if representations differing in the
: order of the terms are considered to be different. In other words
: a(n),n>=1, is the number of positive integral solutions of the
: equation x + y + z = n-1. - Amarnath Murthy
: (amarnath_murthy(AT)yahoo.com), Apr 22 2001
:- is this statement correct?

I think it should be n+2 rather than n-1, or "nonnegative" instead of
"positive". Given a line of n+2 units, C(n+1, 2) is the number of ways
you can choose two gaps to make into plus signs. The first solution,
which yields a(1), must be either 1+1+1=3 or 0+0+0=0.

: Each term, except for the initial 0, is a sum of digits of terms in
: A007908. - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 01 2007
:- obfuscation of T(n)=sum(k=1,n,k),  delete?

It's worse than that, it is wrong after a(9). Definite delete.

: Number of n-permutations of 2 objects u,v, with repetition allowed,
: containing exactly two u's. Example: n=4, a(4) = 6 because we have
: uuvv, uvuv, vuuv, uvvu, vuvu and vvuu. - Zerinvary Lajos
: (zerinvarylajos(AT)yahoo.com), Jul 15 2008
:- I fail to understand. delete?

Essentially: given a list of n-2 v's, the C(n, 2) ways to insert 2 u's.
I'm not sure if finding a clear way to express that would add any value
to A000217.

: Number of units of a(n) belongs to a periodic sequence: 0, 1, 3, 6,
: 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0. [From Mohamed
: Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
:- units apparently means 'least digit' (i.e. mod 10), how to reword?
:  (there appear to be more such statements in the OEIS)

"The sequence is periodic mod 10."? Presumably it is also periodic
mod m for any other modulus, and it would be more useful to mention
the stronger statement.

: Example(a(4)=10): ABCD where A, B, C and D are different links in a
: chain or different amino acids in a peptide possible fragments: A, B,
: C, D, AB, ABC, ABCD, BC, BCD, CD = 10
:- if corresponding comment is kept, move it there, otherwise delete.

Yes, the comment is undecipherable without this example.

:- delete _ALL_BUT_ the following (maple, mathca, pari):
: seq(floor(n^3/(n+1))/2, n=1..25); [From Gary Detlefs
: (gdetlefs(AT)aol.com), Feb 14 2010]

I'd delete this one too (but keep the corresponding formula) - it is
an interesting identity, but I can't see why you'd ever want to use it
to calculate terms.

I notice the sequence has also somehow acquired the 'new' keyword, which
seems very wrong.

Other typos:
- "non-singuliar" -> "non-singular" (comment Benoit Cloitre Aug 29 2002)
- "in given by" -> "is given by" (comment Jon Perry Jun 11 2003)

The comment from Xavier Acloque Oct 31 2003 might be rewritten as:
  Centered k-gonal numbers are k a(n) + 1. E.g. ... (as before)

The comment from Noah Priluck Apr 30 2007 might be rewritten as:
  The total number of triangles visible when n-1 cevians are drawn from
the same vertex on a triangle. E.g. with 1 cevian drawn there are 3
triangles visible.

(Is "cevian" in common use? I had to look it up.)


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