0,2

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)

n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)

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).

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

For n>3 a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2002

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.

Coefficient of x^2 in (1+x+2x^2)^n. - Michael Somos May 26 2004

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

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

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

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

a(n) = A126890(n,1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

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

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

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]

A002262(a(n)) = n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2009]

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]

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. 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.

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.

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.

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.

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. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.

A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

S. P. Humphries, Home page

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1018

Milan Janjic, Two Enumerative Functions

Yashar Memarian, On the Maximum Number of Vertices of Minimal Embedded Graphs, Oct 13, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]

Barbarel Tres Mil, Binomial Matrix (I), Psychedelic Geometry Blogspot 09/22/09 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 22 2009]

P. Moree, Convoluted convolved Fibonacci numbers

P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Sandifer, E. How Euler Did It

Eric Weisstein's World of Mathematics, Cramer-Euler Paradox.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: A(x) = x*(2-x)/(1-x)^3. a(n)=binomial(n+1, n-1)+binomial(n, n-1).

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

a(-3-n)=a(n). - Michael Somos May 26 2004

a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005

2*a(n) = A008778(n) - A105163(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 15 2005

a(n) = C(3+n, 2)-C(3+n, 1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 09 2005

a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006

a(n)=3a(n-1)-3a(n-2)+a(n-3). - Paul Curtz (bpcrtz(AT)free.fr), Jan 02 2008

Starting (2, 5, 9, 14,...) = binomial transform of (2, 3, 1, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2008

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]

a(n)=binomial(n+2,n)-1=binomial(n+2,2)-1 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 22 2009]

A000096 := n->n*(n+3)/2;

[seq(binomial(n, 2)-n, n=3..55)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

seq((GAMMA(n+3)/GAMMA(n+1)/2)-1, n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2007

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

seq(add((k), k=2..n), n=1..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007

A000096:=z*(-2+z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

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

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]

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]

(PARI) a(n)=n*(n+3)/2

Triangular numbers (A000217) minus one. Cf. A000217, A034856, A000124, A005581-A005584.

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.

A column of triangle A014473.

Cf. A067550.

Adjacent sequences: A000093 A000094 A000095 this_sequence A000097 A000098 A000099

Sequence in context: A075543 A132315 A132336 this_sequence A080956 A132337 A134189

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000