%I A000096 M1356 N0522
%S A000096 0,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%T A000096 252,275,299,324,350,377,405,434,464,495,527,560,594,629,665,702,740,
%U A000096 779,819,860,902,945,989,1034,1080,1127,1175,1224,1274,1325,1377,1430
%N A000096 n(n+3)/2.
%C A000096 For n >= 1, a(n) = maximal number of pieces that can be obtained by cutting
an annulus with n cuts. - Robert G. Wilson v (rgwv(AT)rgwv.com)
%C A000096 n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P.
Hatzipolakis (xpolakis(AT)otenet.gr)
%C A000096 n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation
of the symmetric group S_n (cf. James and Kerber, Appendix 1).
%C A000096 a(n) is also the multiplicity of the eigenvalue (-2) of the triangle
graph Delta(n+1). (See p. 19 in Biggs). - Felix Goldberg (felixg(AT)tx.technion.ac.il),
Nov 25 2001
%C A000096 For n>3 a(n-3) = dimension of the traveling salesman polytope T(n). -
Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2002
%C A000096 Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172.
- Jon Wild (wild(AT)music.mcgill.ca), May 07 2004.
%C A000096 Coefficient of x^2 in (1+x+2x^2)^n. - Michael Somos May 26 2004
%C A000096 A curve of order n is generally determined by n(n+3)/2 points. This function
is semiprime for n = 3, 4, 7, 10, 11, 14, 19, 23, 26, 31, 34, 38,
43, ... - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 25 2005
%C A000096 a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomnio
cannot be formed by connecting any other n-polyominoes except for
the n-monomino and the n-monomino is not prime. E.g. for n=1, the
1-monomino is the line of length 1 and the only "prime" 1-polyominoes
are the lines of length 2 and 3. This refers to "free" n-dimensional
polyominoes, i.e. that can be rotated along any axis. - Bryan Jacobs
(bryanjj(AT)gmail.com), Apr 01 2005
%C A000096 Solutions to the quadratic equation q(m, r) = (-3 +/- sqrt(9 + 8(m -
r))) / 2, where m - r is included in a(n). Let t(m) = the triangle
number (A000217) less than some number k and r = k - t(m). If k is
neither prime nor a power of two and m - r is included in A000096,
then m - q(m, r) will produce a value that shares a divisor with
k. - Andrew Plewe, Jun 18 2005
%C A000096 Sum[4/(k*(k+1)*(k-1)),{k,2,n+1}] = ((n+3)*n)/((n+2)*(n+1)). Numerator[Sum[4/
(k*(k+1)*(k-1)),{k,2,n+1}] = (n+3)*n/2 - Alexander Adamchuk (alex(AT)kolmogorov.com),
Apr 11 2006
%C A000096 a(n) = A126890(n,1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Dec 30 2006
%C A000096 Number of rooted trees with n+3 nodes of valence 1, no nodes of valence
2 and exactly two other nodes. I.e. number of planted trees with
n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu),
Jun 10 2007
%C A000096 If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to
the number of (n-2)-subsets of X intersecting Y. > - Milan R. Janjic
(agnus(AT)blic.net), Jul 30 2007
%C A000096 For n>=1, a(n) is the number of distinct shuffles of the identity permutation
on n+1 letters with the identity permutation on 2 letters (12). [From
Camillia Smith (cammie(AT)math.harvard.edu), Oct 04 2008]
%C A000096 A002262(a(n)) = n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
May 20 2009]
%C A000096 Theorem 2, p. 3, of Yashar Memarian, states "let G be a 4-regular minimal
graph on the plane with n attaching points. Then G has at most (n/
2)C2 + n vertices if n is even, else 0. This is sharp. For each n,
there is a minimal 4-regular graph which achieves this bound." Since
xC2 = (1/2)*(x^2) - (1/2)x, the expression (n/2)C2 + n simplifies
to (1/8)*(x^2) + (3/4)*x which is n(n+3)/2 for n an even value of
x. Hence I'd say: "let G be a 4-regular minimal graph on the plane
with n attaching points. Then G has at most A000096(n) = n(n+3)/2
vertices if n is even, else 0. This is sharp." [From Jonathan Vos
Post (jvospost3(AT)gmail.com), Oct 14 2009]
%D A000096 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 797.
%D A000096 D. Applegate, R. Bixby, V. Chvatal and W. Cook : On the solution of traveling
salesman problem. In : Int. Congress of mathematics (Berlin 1998),
Documenta Math., Extra Volume ICM 1998, Vol. III, pp. 645-656.
%D A000096 Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press,
1993.
%D A000096 Euler, L. "Sur une contradiction apparente dans la doctrine des lignes
courbes." Memoires de l'Academie des Sciences de Berlin, 4, 219-233,
1750 Reprinted in Opera Omnia, Series I, Vol. 26. pp. 33-45.
%D A000096 S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and
rings of invariants, Topology and its Applications, 95 (3) (1999)
pp. 173-205.
%D A000096 G. James and A. Kerber, The Representation Theory of the Symmetric Group,
Encyclopedia of Maths. and its Appls., Vol. 16, Addison-Wesley, 1981,
Reading, MA, U.S.A.
%D A000096 D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.
%D A000096 A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag.,
80 (No. 1, 2007), 29-37.
%D A000096 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000096 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A000096 Franklin T. Adams-Watters, <a href="b000096.txt">Table of n, a(n) for
n = 0..10000</a>
%H A000096 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000096 S. P. Humphries, <a href="http://www.math.byu.edu/~steve/">Home page</
a>
%H A000096 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1018">
Encyclopedia of Combinatorial Structures 1018</a>
%H A000096 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000096 Yashar Memarian, <a href="http://arxiv.org/abs/0910.2469">On the Maximum
Number of Vertices of Minimal Embedded Graphs</a>, Oct 13, 2009.
[From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]
%H A000096 Barbarel Tres Mil, <a href="http://psychedelic-geometry.blogspot.com/
2009/09/binomial-matrix-i.html">Binomial Matrix (I)</a>, Psychedelic
Geometry Blogspot 09/22/09 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es),
Sep 22 2009]
%H A000096 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved
Fibonacci numbers</a>
%H A000096 P. Moree, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Convoluted
Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol.
7 (2004), Article 04.2.2.
%H A000096 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000096 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000096 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions,
Explorations and Formulas of the Euler/Pascal Cube</a>.
%H A000096 Sandifer, E. <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2010%20Cramers%20Paradox.pdf"\
>How Euler Did It</a>
%H A000096 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Cramer-EulerParadox.html">Cramer-Euler Paradox</a>.
%H A000096 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000096 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000096 G.f.: A(x) = x*(2-x)/(1-x)^3. a(n)=binomial(n+1, n-1)+binomial(n, n-1).
%F A000096 a(n)=2*t(n)-t(n-1), e.g. a(5)=2*t(5)-t(4)=2*15-10=20. - Jon Perry (perry(AT)globalnet.co.uk),
Jul 23 2003
%F A000096 a(-3-n)=a(n). - Michael Somos May 26 2004
%F A000096 a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01
2005
%F A000096 2*a(n) = A008778(n) - A105163(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Apr 15 2005
%F A000096 a(n) = C(3+n, 2)-C(3+n, 1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 09 2005
%F A000096 a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk (alex(AT)kolmogorov.com),
May 20 2006
%F A000096 a(n)=3a(n-1)-3a(n-2)+a(n-3). - Paul Curtz (bpcrtz(AT)free.fr), Jan 02
2008
%F A000096 Starting (2, 5, 9, 14,...) = binomial transform of (2, 3, 1, 0, 0, 0,
...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2008
%F A000096 For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586.
[From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 27 2009]
%F A000096 a(n)=binomial(n+2,n)-1=binomial(n+2,2)-1 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es),
Sep 22 2009]
%p A000096 A000096 := n->n*(n+3)/2;
%p A000096 [seq(binomial(n,2)-n,n=3..55)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Nov 25 2006
%p A000096 seq((GAMMA(n+3)/GAMMA(n+1)/2)-1,n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 23 2007
%p A000096 seq(sum(mul(gcd(k+2,j),j=0..n), k=0..n), n=-1..51); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 01 2007
%p A000096 seq(add((k), k=2..n), n=1..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Sep 14 2007
%p A000096 A000096:=z*(-2+z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A000096 a:=n->sum(numer (k/(k+3)), k=2..n): seq(a(n), n=1..53); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), May 31 2008
%p A000096 a:=n->sum(2+sum(1, k=1..n), k=2..n)/2:seq(a(n), n=1...43); [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A000096 lst={};Do[AppendTo[lst, n*(n+3)/2], {n, 0, 5!}];lst ...and/or... s=0;
lst={s};Do[s+=n+1;AppendTo[lst, s], {n, 1, 5!}];lst [From Vladimir
Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]
%o A000096 (PARI) a(n)=n*(n+3)/2
%Y A000096 Triangular numbers (A000217) minus one. Cf. A000217, A034856, A000124,
A005581-A005584.
%Y A000096 Occurs as a diagonal in A074079/A074080, i.e.: A074079(n+3, n) = A000096(n-1)
for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2.
- Antti Karttunen, Aug 20, 2002.
%Y A000096 A column of triangle A014473.
%Y A000096 Cf. A067550.
%Y A000096 Sequence in context: A075543 A132315 A132336 this_sequence A080956 A132337
A134189
%Y A000096 Adjacent sequences: A000093 A000094 A000095 this_sequence A000097 A000098
A000099
%K A000096 nonn,easy,nice
%O A000096 0,2
%A A000096 N. J. A. Sloane (njas(AT)research.att.com).
%E A000096 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000
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