%I A000124 M1041 N0391
%S A000124 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,
%T A000124 232,254,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,
%U A000124 742,781,821,862,904,947,991,1036,1082,1129,1177,1226,1276,1327,1379
%N A000124 Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1;
or, maximal number of pieces formed when slicing a pancake with n
cuts.
%C A000124 These are Hogben's central polygonal numbers with the (two-dimensioanl)
symbol
%C A000124 2
%C A000124 .P
%C A000124 1 n
%C A000124 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.
%C A000124 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.
%C A000124 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
%C A000124 Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric
Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2002
%C A000124 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
%C A000124 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
%C A000124 a(n) = A108561(n+3,2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 10 2005
%C A000124 Number of interval subsets of {1,2,3,...,n} (cf. A002662). - Jose Luis
Arregui (arregui(AT)unizar.es), Jun 27 2006
%C A000124 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
%C A000124 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
%C A000124 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
%C A000124 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
%C A000124 Equals row sums of triangle A144328 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 18 2008]
%C A000124 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]
%C A000124 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]
%C A000124 Contribution from Srikanth K S (sriperso(AT)gmail.com), Oct 22 2009:
(Start)
%C A000124 For n\ge 2, a(n) gives the number of sets of subsets $A_1,A_2,\dots A_n$
%C A000124 of $[n]=\{1,2,\dots ,n\}$ so that $\cap_{i=1}^{n} A_i=\emptyset$ and
the sum
%C A000124 $\sum_{\forall j\in [n]}\left (|\cap_{i=1,i\ne j}^{n} A_i|\right )$ is
maximum (End)
%D A000124 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in
Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
%D A000124 A. Burstein and T. Mansour, Words restricted by 3-letter ..., Annals.
Combin., 7 (2003), 1-14; see Example 3.5.
%D A000124 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D A000124 H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page
177.
%D A000124 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer
Press, NY, 1950, p. 22.
%D A000124 Clark Kimberling, Complementary Equations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.4.
%D A000124 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965),
292-298.
%D A000124 D. J. Price, Some unusual series occurring in n-dimensional geometry,
Math. Gaz., 30 (1946), 149-150.
%D A000124 N. Reading, On the structure of Bruhat Order, Ph.D. dissertation, University
of Minnesota, anticipated 2002.
%D A000124 N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties
for Posets, Order, Vol. 19, no. 1 (2002), 73-100.
%D A000124 A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p.
213.
%D A000124 R. Simion and F.W. Schmidt, Restricted Permutations, Europ. J. Comb.,
6, 1985, 383-406.
%D A000124 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000124 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.
%D A000124 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000124 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945,
p. 30.
%D A000124 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)
%H A000124 T. D. Noe, <a href="b000124.txt">Table of n, a(n) for n = 0..1000</a>
%H A000124 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">
The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%H A000124 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000124 H. Bottomley, <a href="A000124.gif">Illustration of initial terms</a>
%H A000124 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">
Words restricted by 3-letter ...</a>.
%H A000124 David Coles, <a href="http://davcoles.tripod.com">Triangle Puzzle</a>
.
%H A000124 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=386">
Encyclopedia of Combinatorial Structures 386</a>
%H A000124 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Partial Complements and Transposable Dispersions</
a>, J. Integer Seqs., Vol. 7, 2004.
%H A000124 Jim Loy, <a href="http://www.jimloy.com/puzz/cole.htm">Triangle Puzzle</
a>.
%H A000124 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations
avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in
S_4</a>
%H A000124 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 A000124 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 A000124 N. Reading, <a href="http://www.math.umn.edu/~reading/dissective.ps">
Order Dimension, Strong Bruhat Order and Lattice Properties for Posets</
a>
%H A000124 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">
On single-deletion-correcting codes</a>
%H A000124 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CircleDivisionbyLines.html">Link to a section of The World of Mathematics
(1).</a>
%H A000124 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PlaneDivisionbyLines.html">Link to a section of The World of Mathematics
(2).</a>
%H A000124 Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http:/
/www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</
a>, vol. 8 (2008).
%H A000124 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000124 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related
to centered polygonal numbers</a>
%H A000124 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000124 G.f.: (1-x+x^2)/(1-x)^3. Equals a triangular number plus 1.
%F A000124 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]
%F A000124 a(n)=sum{k=0..n+1, binomial(n+1, 2(k-n))} - Paul Barry (pbarry(AT)wit.ie),
Aug 29 2004
%F A000124 Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - Michael
Somos Sep 04 2006
%F A000124 G.f.: (1-x^6)/((1-x)^2*(1-x^2)*(1-x^3)). a(-1-n)=a(n). - Michael Somos
Sep 04 2006
%F A000124 binomial(n+2,1)-2*binomial(n+1,1)+binomial(n+2,2). - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), May 12 2006
%F A000124 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
%F A000124 a(n) = A086601(n)^(1/2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 25 2008
%F A000124 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]
%F A000124 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:
%F A000124 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
%F A000124 delta(l_1,l_2,...,l_i,...,l_n)
%F A000124 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i <> l_(i+1) and l_(i+1)
<> 0
%F A000124 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
%F A000124 a(n) = A000217(n) - (n-1) for n >= 2. A000217(n) = triangular numbers.
[From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 16 2009]
%F A000124 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]
%e A000124 a(3)=7 because the 132- and 321-avoiding permutations of {1,2,3,4} are
1234,2134,3124,2314,4123,3412,2341.
%p A000124 A000124 := n-> n*(n+1)/2+1;
%p A000124 A000124:=-(1-z+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A000124 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
%p A000124 with (combinat):seq((fibonacci(3, n)+n+1)/2, n=0..52); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 07 2008
%p A000124 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(Carbo\
n, 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]
%t A000124 Table[(Binomial[i+2, 2]+1),{i,-1, 51}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Mar 23 2007
%t A000124 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]
%t A000124 ...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]
%o A000124 (PARI) {a(n)=(n^2+n)/2+1} /* Michael Somos Sep 04 2006 */
%Y A000124 A000124 = triangular numbers A000217(n)+1. Partial sums =(A033547)/2,
(A014206)/2. Cf. A000125, A003600, A016028, A000096, A055503, A002061.
%Y A000124 The first 20 terms are also found in A025732 and A025739.
%Y A000124 Cf. A002522, A072863, A144328.
%Y A000124 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]
%Y A000124 Sequence in context: A025725 A025732 A025739 this_sequence A152947 A098574
A005689
%Y A000124 Adjacent sequences: A000121 A000122 A000123 this_sequence A000125 A000126
A000127
%K A000124 easy,core,nonn,nice
%O A000124 0,2
%A A000124 N. J. A. Sloane (njas(AT)research.att.com).
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