0,2

These are Hogben's central polygonal numbers with the (two-dimensioanl) symbol

2

.P

1 n

m=(n-1)(n-2)/2+1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. - Keith M. Briggs, May 14 2004.

Also maximal number of grandchildren of a binary vector of length n+2. E.g. a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.

This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002

Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2002

For n>=1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n) - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003

Narayana transform (analogue of the binomial transform) of vector [1, 1, 0, 0, 0...] = A000124; using the infinite lower Narayana triangle of A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0...] = A000124. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2005

a(n) = A108561(n+3,2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005

Number of interval subsets of {1,2,3,...,n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006

Define a number of straight lines in the plane to be in general arrangement when (1) no two lines are parallel, (2) there is no point common to three lines. Then these are the maximal numbers of regions defined by n straight lines in general arrangement in the plane. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

Note that a(n) = a(n-1) + A000027(n-1). This has the following geometrical interpretation: Suppose there are already n-1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n-1 lines, and now one more line is added in general arrangement. Then it will cut each of the n-1 lines and acquire intersection points which are in general arrangement. (See the comments on A000027 for general arrangement with points.) These points on the new line define the maximal number of regions in 1-space definable by n-1 points, hence this is A000027(n-1), where for A000027 an offset of 0 is assumed, that is, A000027(n-1)=(n+1)-1=n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n)=a(n-1)+A000027(n-1). Cf. the comments on A000125 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the n-th element of this sequence is the number of edges in the n-th zone added with the n-th "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron). E.g. adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. - Shel Kaphan (skaphan(AT)gmail.com), Feb 16 2006

If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of (n-2)-subsets of X which have no exactly one element in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 28 2007

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.

A. Burstein and T. Mansour, Words restricted by 3-letter ..., Annals. Combin., 7 (2003), 1-14; see Example 3.5.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.

H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.

N. Reading, On the structure of Bruhat Order, Ph.D. dissertation, University of Minnesota, anticipated 2002.

N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets, Order, Vol. 19, no. 1 (2002), 73-100.

A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.

R. Simion and F.W. Schmidt, Restricted Permutations, Europ. J. Comb., 6, 1985, 383-406.

N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.

A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964)

R. B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventues in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.

T. D. Noe, Table of n, a(n) for n = 0..1000

Milan Janjic, Two Enumerative Functions

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.

H. Bottomley, Illustration of initial terms

A. Burstein and T. Mansour, Words restricted by 3-letter ....

David Coles, Triangle Puzzle.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 386

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Jim Loy, Triangle Puzzle.

T. Mansour, Permutations avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in S_4

N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets

N. J. A. Sloane, On single-deletion-correcting codes

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

Index entries for "core" sequences

Index entries for sequences related to centered polygonal numbers

G.f.: (1-x+x^2)/(1-x)^3. Equals a triangular number plus 1.

a(n)=sum{k=0..n+1, binomial(n+1, 2(k-n))} - Paul Barry (pbarry(AT)wit.ie), Aug 29 2004

Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - Michael Somos Sep 04 2006

G.f.: (1-x^6)/((1-x)^2*(1-x^2)*(1-x^3)). a(-1-n)=a(n). - Michael Somos Sep 04 2006

binomial(n+2,1)-2*binomial(n+1,1)+binomial(n+2,2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006

Binomial transform of (1, 1, 1, 0, 0, 0,...) and inverse binomial transform of A072863: (1, 3, 9, 26, 72, 192,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 15 2007

a(n) = A086601(n)^(1/2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2008

a(3)=7 because the 132- and 321-avoiding permutations of {1,2,3,4} are 1234,2134,3124,2314,4123,3412,2341.

A000124 := n-> n*(n+1)/2+1;

A000124:=-(1-z+z**2)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]

with (combinat):a:=n->sum(fibonacci(4, i), i=0..n): seq(sqrt(a(n)+1), n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2008

Table[(Binomial[i+2, 2]+1), {i, -1, 51}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 23 2007

(PARI) {a(n)=(n^2+n)/2+1} /* Michael Somos Sep 04 2006 */

A000124 = triangular numbers A000217(n)+1. Partial sums =(A033547)/2, (A014206)/2. Cf. A000125, A003600, A016028, A000096, A055503, A002061.

The first 20 terms are also found in A025732 and A025739.

Cf. A072863.

Adjacent sequences: A000121 A000122 A000123 this_sequence A000125 A000126 A000127

Sequence in context: A025725 A025732 A025739 this_sequence A098574 A005689 A131075

easy,core,nonn,nice,new

njas