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A000124
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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.
(Formerly M1041 N0391)
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221
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1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379
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OFFSET
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0,2
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COMMENTS
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These are Hogben's central polygonal numbers with the (two-dimensional) symbol
2
.P
1 n
The first line cuts the pancake into 2 pieces. For n>1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2+3+4+5+ ... +16 = 137. These are the triangular numbers plus 1 (cf. A000217).
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, 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
Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see A000140.
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, Apr 28 2005
a(n) = A108561(n+3,2). - Reinhard Zumkeller, 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, 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 Janjic, Dec 28 2007
Equals row sums of triangle A144328 [From Gary W. Adamson, 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 i-th Fibonacci number. [From John W. Layman, Dec 02 2008]
a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. [From Geoffrey Critzer, Mar 10 2009]
For n >= 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. [From Srikanth K S, Oct 22 2009]
The numbers along the left edge of Floyd's triangle. [From Paul Muljadi, Jan 25 2010]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x). [From Milan Janjic, Jan 24 2010]
Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. [From Johannes W. Meijer, June 21, 2010]
(1 + x^2 + x^3 + x^4 + x^5 + ...)*(1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...) = (1 + 2x + 4x^2 + 7x^3 + 11x^4 + ...). [Gary W. Adamson, Jul 27 2010]
The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. [From Jeffrey Liese, Dec 23 2010]
Let b(0)=b(1)=1; b(n)=max(b(n-1)+n-1,b(n-2)+n-2) then a(n)=b(n+1). [Yalcin Aktar, Jul 28 2011]
Also number of triangular numbers so far, for n > 0: a(n) = a(n-1) + Sum(A010054(a(k)): 0 <= k < n), see also A097602, A131073. [Reinhard Zumkeller, Nov 15 2012]
Also number of unique sums of 1 through n where each of those can be + or -. E.g. {1+2,1-2,-1+2,-1-2}={3,-1,1,-3} and a(2)=4 [Toby Gottfried, Nov 17, 2011]
This sequence is complete because the sum of the first n terms is always greater or equal to a(n+1)-1. Consequently, any nonnegative number can be written as a sum of distinct terms of this sequence. See A204009, A072638. [Frank M Jackson, Jan 09, 2012]
The sequence is the number of distinct sums of subsets of the non-negative integers, and its first differences are the positive integers. See A208531 for similar results for the squares. [John W. Layman, Feb 28 2012]
a(n) = A014132(n,1) for n > 0. - Reinhard Zumkeller, Dec 12 2012
Apparently the number of Dyck paths of semilength n+1 in which the sum of the first and second ascents add to n+1. - David Scambler, Apr 22 2013
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REFERENCES
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R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
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.
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83. [From Robert G. Wilson v, May 21 2010]
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)
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LINKS
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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)
H. Bottomley, Illustration of initial terms
A. Burstein and T. Mansour, Words restricted by 3-letter ....
David Coles, Triangle Puzzle.
Guo-Niu Han, Enumeration of Standard Puzzles
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 386
Milan Janjic, Two Enumerative Functions
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
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.
_Simon 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.
_Simon 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, Circle Division by Lines
Eric Weisstein's World of Mathematics, Plane Division by Lines
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Wikipedia, Floyd's triangle [From Paul Muljadi, Jan 25 2010]
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, signature (3,-3,1).
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FORMULA
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G.f.: (1-x+x^2)/(1-x)^3.
G.f.: (1-x^6)/((1-x)^2*(1-x^2)*(1-x^3)). a(-1-n)=a(n). - Michael Somos Sep 04 2006
a(n+3)=3*a(n+2)-3*a(n+1)+a(n) and a(1)=1, a(2)=2, a(3)=4. [From Artur Jasinski, Oct 21 2008]
a(n) = A000217(n)+1.
a(n)=a(n-1)+n. E.g.f.:(1+x+x^2/2)*exp(x) [From Geoffrey Critzer, Mar 10 2009]
a(n)=sum{k=0..n+1, binomial(n+1, 2(k-n))} - Paul Barry, Aug 29 2004
Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - Michael Somos, Sep 04 2006
binomial(n+2,1)-2*binomial(n+1,1)+binomial(n+2,2). - Zerinvary Lajos, 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, Oct 15 2007
a(n) = A086601(n)^(1/2). - Zerinvary Lajos, Apr 25 2008
From Thomas Wieder, Feb 25 2009: (Start)
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. (End)
a(n) = A034856(n+1) - A005843(n) = A000217(n) + A005408(n) - A005843(n). [From Jaroslav Krizek, Sep 05 2009]
a(n) = 2*a(n-1)-a(n-2)+1. [From Eric Werley, Jun 27 2011]
E.g.f.: exp(x)*(1+x+(x^2)/2)=Q(0); Q(k)=1+x/(1-x/(2+x-4/(2+x*(k+1)/Q(k+1)))) ; (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
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EXAMPLE
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a(3)=7 because the 132- and 321-avoiding permutations of {1,2,3,4} are 1234,2134,3124,2314,4123,3412,2341.
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MAPLE
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A000124 := n-> n*(n+1)/2+1;
A000124:=-(1-z+z**2)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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FoldList[#1 + #2 &, 1, Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[Range[0, 60]]+1 (* Harvey P. Dale, Mar 12 2013 *)
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PROG
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(PARI) {a(n)=(n^2+n)/2+1} /* Michael Somos, Sep 04 2006 */
(Haskell)
a000124 = (+ 1) . a000217
-- Reinhard Zumkeller, Oct 04 2012, Nov 15 2011
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CROSSREFS
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Slicing a cake: A000125, a bagel: A003600.
Partial sums =(A033547)/2, (A014206)/2.
Cf. A016028, A000096, A055503, A002061.
The first 20 terms are also found in A025732 and A025739.
Cf. A002522, A072863, A144328.
Cf. A005408, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller, Jun 17 2009]
Cf. A014206. [From Robert G. Wilson v, May 21 2010]
Sequence in context: A025732 A025739 * A152947 A212369 A212368 A217838
Adjacent sequences: A000121 A000122 A000123 * A000125 A000126 A000127
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KEYWORD
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easy,core,nonn,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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