

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)


427



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

0,2


COMMENTS

These are Hogben's central polygonal numbers with the (twodimensional) symbol
2
.P
1 n
The first line cuts the pancake into 2 pieces. For n > 1, the nth 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 = (n1)(n2)/2 + 1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected.  Keith 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 321avoiding 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 (analog 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
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(n1) + A000027(n1). This has the following geometrical interpretation: Suppose there are already n1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n1 lines and now one more line is added in general arrangement. Then it will cut each of the n1 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 1space definable by n1 points, hence this is A000027(n1), where for A000027 an offset of 0 is assumed, that is, A000027(n1) = (n+1)1 = n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n1) regions already there, hence a(n) = a(n1) + A000027(n1). 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) 3d nonintersecting parallelepipeds, the nth element of this sequence is the number of edges in the nth zone added with the nth "layer" of parallelepipeds. (Verified up to 10zone 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 5valence vertex and counting visible vertices.  Shel Kaphan, Feb 16 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 2subset of an nset X then, for n >= 3, a(n3) is the number of (n2)subsets of X which do not have exactly one element in common with Y.  Milan Janjic, Dec 28 2007
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.  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, ..., A_n of n = {1, 2, ..., n} such that Meet_{i = 1..n} A_i is empty and Sum_{j in [n]} (Meet{i = 1..n, i != j} A_i) is a 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,i1] = 1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n1) = (1)^(n1)*coeff(charpoly(A,x),x).  Milan Janjic, Jan 24 2010
Also the number of deck entries of Euler's ship. See the MeijerNepveu 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 0digits between any pair of consecutive 1digits.  Jeffrey Liese, Dec 23 2010
Let b(0) = b(1) = 1; b(n) = max(b(n1)+n1, b(n2)+n2) then a(n) = b(n+1).  Yalcin Aktar, Jul 28 2011
Also number of distinct sums of 1 through n where each of those can be + or . E.g., {1+2,12,1+2,12} = {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 than 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 nonnegative integers, and its first differences are the positive integers. See A208531 for similar results for the squares.  John W. Layman, Feb 28 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 nth partial sum of sequence 1, 1, 2. Explanation: 1st partial sums of 1, 1, 2 are 1, 2, 4; 2nd partial sums are 1, 3, 7; 3rd partial sums are 1, 4, 11; 4th partial sums are 1, 5, 16, etc.  Bob Selcoe, Jul 04 2013
Equivalently, numbers of the form 2*m^2+m+1, where m = 0, 1, 1, 2, 2, 3, 3, ... .  Bruno Berselli, Apr 08 2014
For n >= 2: quasitriangular numbers; the almosttriangular numbers being A000096(n), n >= 2. Note that 2 is simultaneously almosttriangular and quasitriangular.  Daniel Forgues, Apr 21 2015
n points in general position determine "n choose 2" lines, so A055503(n) <= a(n(n1)/2). If n > 3, the lines are not in general position and so A055503(n) < a(n(n1)/2).  Jonathan Sondow, Dec 01 2015
The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10gonal numbers (A062786), for n > 0, A133292.  Peter M. Chema, Sep 15 2016
For n >= 0, a(n) is the number of weakly unimodal sequences of length n over the alphabet {1, 2}.  Armend Shabani, Mar 10 2017
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) != e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) and e(i) < e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) != e(k). [Martinez and Savage, 2.4]
(End)
The odd prime factors != 7 occur in an interval of p successive terms either never or exactly twice, while 7 always occurs only once. If a prime factor p appears in a(n) and a(m) within such an interval, then n + m == 1 (mod p). When 7 divides a(n), then 2*n == 1 (mod 7). a(n) is never divisible by the prime numbers given in A003625.
While all prime factors p != 7 can occur to any power, a(n) is never divisible by 7^2. The prime factors are given in A045373. The prime terms of this sequence are given in A055469.
(End)
a(n1) is the greatest sum of arch lengths for the top arches of a semimeander with n arches. An arch length is the number of arches covered + 1.
/\ The top arch has a length of 3. /\ The top arch has a length of 3.
/ \ Both bottom arches have a //\\ The middle arch has a length of 2.
//\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
Example: for n = 4, a(41) = a(3) = 7 /\
//\\
/\ ///\\\ 1 + 3 + 2 + 1 = 7. (End)
a(n+1) is the a(n)th smallest positive integer that has not yet appeared in the sequence.  Matthew Malone, Aug 26 2021
For n> 0, let the ndimensional cube {0,1}^n be, provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 2. Example: n = 4. (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0) are at distance <= 2 from (0,0,0,0), so a(4) = 11.  Yosu Yurramendi, Dec 10 2021
a(n) is the sum of the first three entries of row n of Pascal's triangle.  Daniel T. Martin, Apr 13 2022
a(n1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 3 with exactly one descent. For example, sigma is one of the patterns, {132, 213, 231, 312}.  Jessica A. Tomasko, Sep 14 2022
a(n+4) is the number of ways to tile an equilateral triangle of side length 2*n with smaller equilateral triangles of side length n and side length 1. For example, with n=2, there are 22 ways to tile an equilateral triangle of side length 4 with smaller ones of sides 2 and 1, including the one tiling with sixteen triangles of sides 1 and the one tiling with four triangles of sides 2.  Ahmed ElKhatib and Greg Dresden, Aug 19 2024


REFERENCES

Robert B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
Henry Ernest Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence  see "List of Sequences" in Vol. 2.
Alain M. Robert, A Course in padic Analysis, SpringerVerlag, 2000; p. 213.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On singledeletioncorrecting codes, in Codes and Designs (Columbus, OH, 2000), 273291, 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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 98.
William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
Akiva M. Yaglom and Isaak 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: HoldenDay, Inc., 1964).


LINKS

M. F. Hasler, Interactive illustration of A000124. [Sep 06 2017: The user can choose the slices to make, but the program can suggest a set of n slices which should yield the maximum number of pieces. For n slices this obviously requires 2n endpoints, or 2n+1 if they are equally spaced, so if there are not enough "blobs", their number is accordingly increased. This is the distinction between "draw" (done when you change the slices or number of blobs by hand) and "suggest" (propose a new set of slices).]
Rodica Simion and Frank W. Schmidt, Restricted permutations, European J. Combin., 6, 383406, 1985; see Example 3.5.


FORMULA

G.f.: (1  x + x^2)/(1  x)^3.  Simon Plouffe in his 1992 dissertation
G.f.: (1  x^6)/((1  x)^2*(1  x^2)*(1  x^3)). a(n) = a(1  n) for all n in Z.  Michael Somos, Sep 04 2006
Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, 1].  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) = a(n1) + 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
a(n) = binomial(n+2, 1)  2*binomial(n+1, 1) + binomial(n+2, 2).  Zerinvary Lajos, May 12 2006
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) = 2*a(n1)  a(n2) + 1.  Eric Werley, Jun 27 2011
E.g.f.: exp(x)*(1+x+(x^2)/2) = Q(0); Q(k) = 1+x/(1x/(2+x4/(2+x*(k+1)/Q(k+1)))); (continued fraction).  Sergei N. Gladkovskii, Nov 21 2011
Dirichlet g.f.: (zeta(s2) + zeta(s1) + 2*zeta(s))/2.
Sum_{n >= 0} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = A226985. (End)
a(n) = 2*a(n1)  binomial(n1, 2) and a(0) = 1.  Armend Shabani, Mar 10 2017
a(n) = (Sum_{i=n2..n+2} A000217(i))/5.
a(n) = (Sum_{i=n2..n+2} A002378(i))/10.
a(n) = (Sum_{i=n..n+2} A002061(i)+1)/6.
a(n) = (Sum_{i=n1..n+2} A000290(i)+2)/8.
(End)
Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(15)*Pi/2)*sech(sqrt(7)*Pi/2).
Product_{n>=1} (1  1/a(n)) = 2*Pi*sech(sqrt(7)*Pi/2). (End)
a((n^23n+6)/2) + a((n^2n+4)/2) = a(n^22n+6)/2.  Charlie Marion, Feb 14 2023


EXAMPLE

a(3) = 7 because the 132 and 321avoiding permutations of {1, 2, 3, 4} are 1234, 2134, 3124, 2314, 4123, 3412, 2341.
G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 22*x^6 + 29*x^7 + ...


MAPLE



MATHEMATICA

Table[PolygonalNumber[n] + 1, {n, 0, 52}] (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)


PROG

(PARI) {a(n) = (n^2 + n) / 2 + 1}; /* Michael Somos, Sep 04 2006 */
(Haskell)
a000124 = (+ 1) . a000217
(Python)
def a(n): return n*(n+1)//2 + 1


CROSSREFS

Cf. A000096 (Maximal number of pieces that can be obtained by cutting an annulus with n cuts, for n >= 1).
Cf. A002061, A002522, A016028, A055503, A072863, A144328, A177862, A263883, A000127, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A051601, A228918.
Cf. A055469 Quasitriangular primes.


KEYWORD

nonn,core,easy,nice,changed


AUTHOR



STATUS

approved



