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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.
(Formerly M1041 N0391)
258
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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. - 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. - John W. Layman, Dec 02 2008

a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. - 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. - Srikanth K S, Oct 22 2009

The numbers along the left edge of Floyd's triangle. - 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). - Milan Janjic, Jan 24 2010

Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. - Johannes W. Meijer, Jun 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. - 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

Without 1 and 2, a(n) equals the terminus of the n-th partial sum of sequence 1,1,2.  Explanation: 1-st partial sums of 1,1,2 are 1,2,4; 2-nd partial sums are 1,3,7; 3-rd  partial sums are 1,4,11; 4-th partial sums are 1,5,16, etc. - Bob Selcoe, Jul 04 2013

a(n) = A228074(n+1,n). - Reinhard Zumkeller, Aug 15 2013

For n>3, a(n) is the number of length n binary words that have at least two 1's and at most two 0's.  a(4) = 11 because we have: 0011, 0101, 0110, 0111, 1001, 1010, 1011, 1100, 1101, 1110, 1111. - Geoffrey Critzer, Jan 08 2014

For n > 0: A228446(a(n)) = 3. - Reinhard Zumkeller, Mar 12 2014

Equivalently, numbers of the form 2*m^2+m+1, where m = 0,-1,1,-2,2,-3,3,... [Bruno Berselli, Apr 08 2014]

REFERENCES

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

A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014, http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.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.

Markus Moll, On a family of random noble means substitutions, Dr. Math. Dissertation, Universität Bielefeld, 2013; http://pub.uni-bielefeld.de/luur/download?func=downloadFile&recordOId=2637807&fileOId=2637828

Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83. [Robert G. Wilson v, May 21 2010]

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

L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, http://faculty.valpo.edu/lpudwell/slides/pp2014_pudwell.pdf, 2014

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

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)

LINKS

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)

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

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 [Cached copy]

Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657, 2014

C. Homberger, V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946, 2013

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 386 [broken link?]

Milan Janjic, Two Enumerative Functions

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

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

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

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

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

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 [Paul Muljadi, Jan 25 2010]

Index entries for "core" sequences

Index entries for sequences related to centered polygonal numbers

Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

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. - 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). - Geoffrey Critzer, Mar 10 2009

a(n) = sum(k=0..n+1, binomial(n+1, 2(k-n))). - Paul Barry, Aug 29 2004

Binomial(n+2,1)-2*binomial(n+1,1)+binomial(n+2,2). - Zerinvary Lajos, May 12 2006

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). - Jaroslav Krizek, Sep 05 2009

a(n) = 2*a(n-1)-a(n-2)+1. - 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

a(n) = 1 + floor(n/2) + ceiling(n^2/2) = 1 + A004526(n) + A000982(n). - Wesley Ivan Hurt, Jun 14 2013

EXAMPLE

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

MAPLE

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

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

MATHEMATICA

FoldList[#1 + #2 &, 1, Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)

Accumulate[Range[0, 60]]+1 (* Harvey P. Dale, Mar 12 2013 *)

Select[Range[2000], IntegerQ[Sqrt[8 # - 7]] &] (* Vincenzo Librandi, Apr 16 2014 *)

PROG

(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

(MAGMA) [n: n in [0..1500] | IsSquare(8*n-7)]; // Vincenzo Librandi, Apr 16 2014

CROSSREFS

Cf. A000096, A002061, A002522, A016028, A055503, A072863, A144328.

Slicing a cake: A000125, a bagel: A003600.

Partial sums =(A033547)/2, (A014206)/2.

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

Cf. A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A014206, A051601.

Sequence in context: A025732 A025739 * A152947 A212369 A212368 A217838

Adjacent sequences:  A000121 A000122 A000123 * A000125 A000126 A000127

KEYWORD

nonn,core,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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