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)gmail.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

Equals row sums of triangle A144328 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 18 2008]

It appears that a(n) is the number of distinct values among the fractions F(i+1)/F(j+1) as j ranges from 1 to n and, for each fixed j, i ranges from 1 to j, where F(i) denotes the ith Fibonacci number. [From John W. Layman (layman(AT)math.vt.edu), Dec 02 2008]

a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 10 2009]

Contribution from Srikanth K S (sriperso(AT)gmail.com), Oct 22 2009: (Start)

For n\ge 2, a(n) gives the number of sets of subsets $A_1,A_2,\dots A_n$

of $[n]=\{1,2,\dots ,n\}$ so that $\cap_{i=1}^{n} A_i=\emptyset$ and the sum

$\sum_{\forall j\in [n]}\left (|\cap_{i=1,i\ne j}^{n} A_i|\right )$ is maximum (End)

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

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

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.

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

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)

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

David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Milan Janjic, Two Enumerative Functions

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

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.

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

Index entries for sequences related to linear recurrences with constant coefficients

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

a(n)=a(n-1)+n. E.g.f.:(1+x+x^2/2)*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 10 2009]

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

From a(4) recurence formula a(n+3)=3a(n+2)-3a(n+1)+a(n) and a(1)=1, a(2)=2, a(3)=4 (successive powers of two) [From Artur Jasinski (grafix(AT)csl.pl), Oct 21 2008]

Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:

a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}

delta(l_1,l_2,...,l_i,...,l_n)

where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i <> l_(i+1) and l_(i+1) <> 0

and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.

a(n) = A000217(n) - (n-1) for n >= 2. A000217(n) = triangular numbers. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 16 2009]

a(n) = A034856(n+1) - A005843(n) = A000217(n) + A005408(n) - A005843(n) = A000217(n) - 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 05 2009]

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; [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

with (combinat):seq((fibonacci(3, n)+n+1)/2, n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008

with(combstruct); gramm_Alkyl:=Alkyl=Prod(Carbon, Set(Alkyl, card<1)), Carbon=Atom: specs_Alkyl:=[Alkyl, {gramm_Alkyl}, unlabeled]: gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon, S1_Alkyl[X], Set(Alkyl, card<1)), Prod(Prod(Carbon, X), Set(Alkyl, card<1))), X=Epsilon: specs_S1_Alkyl:=[S1_Alkyl[X], {gramm_S1_Alkyl, gramm_Alkyl}, unlabeled]: gramm_S2b_Alkyl:=S2_Alkyl[X, X]=Union(Prod(Carbon, S2_Alkyl[X, X], Set(Alkyl, card<1)), Prod(Carbon, Union(Prod(S1_Alkyl[X], S1_Alkyl[X]), Prod(S1_Alkyl[X], X), Prod(X, X)), Set(Alkyl, card<1))): specs_S2b_Alkyl:=[S2_Alkyl[X, X], {gramm_S2b_Alkyl, gramm_S1_Alkyl, gramm_Alkyl}, unlabeled]: seq(count(specs_S2b_Alkyl, size=i), i=1..53); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 15 2009]

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

a = {k, m, r} = {1, 2, 4}; Do[l = 3 r - 3 m + k; AppendTo[a, l]; k = m; m = r; r = l, {n, 1, 50}]; a [From Artur Jasinski (grafix(AT)csl.pl), Oct 21 2008]

...and/or... i=1; s=1; lst={}; Do[s+=n+i; If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 30 2008]

(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. A002522, A072863, A144328.

A005408, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Sequence in context: A025725 A025732 A025739 this_sequence A152947 A098574 A005689

Adjacent sequences: A000121 A000122 A000123 this_sequence A000125 A000126 A000127

easy,core,nonn,nice

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