[seqfan] editing A000217 Triangular numbers (long message)

[Joerg Arndt]

My questions and remarks regarding A000217,
your comments are most welcome:


(COMMENTS):

 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)


 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?)


 Number of ways two different numbers can be selected from the set
 {0,1,2,...,n} without repetition, or, number of ways two different
 numbers can be selected from the set {1,2,...,n} with repetition.

- second half appears wrong (should be n^2?)
- can I change an existing comment?


 In 24-bit RGB color cube, the number of color-lattice-points in r+g+b
 = n planes at n < 256 equals the triangular numbers. For n = 256,
 ..., 765 the number of legitimate color partitions is less than
 A000217(n) because {r,g,b} components cannot exceed 255. For
 n=256,..,511, the number of non-color partitions are computable with
 A045943(n-255), while for n = 512-765, the number of color points in
 r+g+b planes equals A000217(765-n). - Labos
 E. (labos(AT)ana.sote.hu), Jun 20 2005

- obfuscation by application.  I'd like to delete this one


 Beginning from a(3), a(n) is the number of way to get a semiprime
 from n primes. Example: From 2 and 3 the number of semiprimes is 3:
 2*2, 3*3, 2*3; from 2 and 3 and 5 the number of semiprimes is 6: 2*2,
 3*3, 5*5, 2*3, 2*5, 3*5. - Giovanni Teofilatto
 (g.teofilatto(AT)tiscalinet.it), Sep 17 2006

- obfuscation of (n choose 2), delete?


 Comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 18 2006: (Start)
 a:=n->sum(j + 1,j=-1..n): seq(a(n),n=-1..50);
 a:=n->sum(j + 2,j=0..n): seq(a(n),n=-1..51); => A000096 = this sequence + 1*A001477
 a:=n->sum(j + 2,j=1..n): seq(a(n),n=0..48); => A055998 = this sequence + 2*A001477
 a:=n->sum(j + 2,j=2..n):seq(a(n),n=1..50); => A055999 = this sequence + 3*A001477
 a:=n->sum(j + 2,j=3..n):seq(a(n),n=2..52); => A056000 = this sequence + 4*A001477
 a:=n->sum(j + 2,j=4..n):seq(a(n),n=3..53); => A056115 = this sequence + 5*A001477
 a:=n->sum(j + 2,j=5..n):seq(a(n),n=4..54); => A056119 = this sequence + 6*A001477
 a:=n->sum(j + 2,j=6..n):seq(a(n),n=5..50); => A056121 = this sequence + 7*A001477
 a:=n->sum(j + 2,j=7..n):seq(a(n),n=6..56); => A056126 = this sequence + 8*A001477
 a:=n->sum(j + 2,j=8..n):seq(a(n),n=7..56); => A051942 = this sequence + 9*A001477
 a:=n->sum(j + 2,j=9..n):seq(a(n),n=8..59); => A101859 = this sequence + 10*A001477 (End) 

- gimme a bleeping break.  Delete.


 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.


 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?


 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?


 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?


 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)


 Partial sums of nonnegative numbers. [From Juri-Stepan Gerasimov
 (2stepan(AT)rambler.ru), Jan 25 2010]

- delete (dup with formula)


(LINKS:)

- delete the following:
http://mathworld.wolfram.com/AbsoluteValue.html
http://mathworld.wolfram.com/Composition.html
http://mathworld.wolfram.com/Distance.html
http://mathworld.wolfram.com/GolombRuler.html


(FORMULAS:)

 f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i),
 then a(n)=-f(n,n-1,1), for n>=1. [From Milan R. Janjic
 (agnus(AT)blic.net), Dec 20 2008]

- whatever.  delete


 An exponential generating function for the inverse of this sequence
 is given by sum((pochhammer(1, m)*pochhammer(1,
 m))*x^m/(pochhammer(3, m)*factorial(m)), m = 0
 .. infinity)=((2-2*x)*ln(1-x)+2*x)/x^2; The n-th derivative of which
 has a closed form which must be evaluated by taking the limit
 x=0. A000217[n+1]=limit(Diff(((2-2*x)*ln(1-x)+2*x)/x^2,
 x$n),x=0)^-1=limit((2*GAMMA(n)*(-1/x)^n*(n*(x/(-1+x))^n*(-x+1+n)*LerchPhi(x/(-1+x),
 1,
 n)+(-1+x)*(n+1)*(x/(-1+x))^n+n*(ln(1-x)+ln(-1/(-1+x)))*(-x+1+n))/x^2),x=0)^-1
 [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009]

- WTF?  delete


(EXAMPLE:)

 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.


(CODES:)

- delete _ALL_BUT_ the following (maple, mathca, pari):

 A000217 := proc(n) n*(n+1)/2; end; [ seq(n*(n+1)/2, n=0..100)];

 istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n =
 t1*(t1+1)/2 then RETURN(true) else RETURN(false); fi; end;
 (N. J. A. Sloane, May 25 2008)

 a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+1 od:
 seq(a[n], n=0..50); (Kristof)

 seq(floor(n^3/(n+1))/2, n=1..25); [From Gary Detlefs
 (gdetlefs(AT)aol.com), Feb 14 2010]

 a(n)=n*(n+1)/2


(CROSSREFS:)
 Cf. A000396, A000668. [From Omar E. Pol (info(AT)polprimos.com), Sep
 05 2008]

- delete