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%I M1041 N0391 #636 Jul 22 2023 11:53:50
%S 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,
%T 232,254,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,
%U 742,781,821,862,904,947,991,1036,1082,1129,1177,1226,1276,1327,1379
%N 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 These are Hogben's central polygonal numbers with the (two-dimensional) symbol
%C 2
%C .P
%C 1 n
%C 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).
%C 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 Briggs_, May 14 2004
%C 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 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 Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - _Emeric Deutsch_, Mar 14 2002
%C 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 Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see A000140.
%C 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
%C Number of interval subsets of {1, 2, 3, ..., n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006
%C 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 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 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
%C 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
%C 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 do not have exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007
%C Equals row sums of triangle A144328. - _Gary W. Adamson_, Sep 18 2008
%C 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
%C a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. - _Geoffrey Critzer_, Mar 10 2009
%C 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
%C The numbers along the left edge of Floyd's triangle. - _Paul Muljadi_, Jan 25 2010
%C 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
%C Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. - _Johannes W. Meijer_, Jun 21 2010
%C (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
%C The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. - _Jeffrey Liese_, Dec 23 2010
%C 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
%C 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
%C Also number of distinct 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
%C 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
%C 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
%C 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
%C Without 1 and 2, a(n) equals the terminus of the n-th 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
%C Equivalently, numbers of the form 2*m^2+m+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - _Bruno Berselli_, Apr 08 2014
%C For n >= 2: quasi-triangular numbers; the almost-triangular numbers being A000096(n), n >= 2. Note that 2 is simultaneously almost-triangular and quasi-triangular. - _Daniel Forgues_, Apr 21 2015
%C n points in general position determine "n choose 2" lines, so A055503(n) <= a(n(n-1)/2). If n > 3, the lines are not in general position and so A055503(n) < a(n(n-1)/2). - _Jonathan Sondow_, Dec 01 2015
%C The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10-gonal numbers (A062786), for n > 0, A133292. - _Peter M. Chema_, Sep 15 2016
%C Partial sums of A028310. - _J. Conrad_, Oct 31 2016
%C 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
%C From _Eric M. Schmidt_, Jul 17 2017: (Start)
%C 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]
%C 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]
%C 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]
%C (End)
%C Numbers m such that 8m - 7 is a square. - _Bruce J. Nicholson_, Jul 24 2017
%C From _Klaus Purath_, Jan 29 2020: (Start)
%C 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.
%C 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.
%C (End)
%C From _Roger Ford_, May 10 2021: (Start)
%C a(n-1) is the greatest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
%C /\ The top arch has a length of 3. /\ The top arch has a length of 3.
%C / \ Both bottom arches have a //\\ The middle arch has a length of 2.
%C //\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
%C Example: for n = 4, a(4-1) = a(3) = 7 /\
%C //\\
%C /\ ///\\\ 1 + 3 + 2 + 1 = 7. (End)
%C a(n+1) is the a(n)-th smallest positive integer that has not yet appeared in the sequence. - _Matthew Malone_, Aug 26 2021
%C For n> 0, let the n-dimensional 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
%C a(n) is the sum of the first three entries of row n of Pascal's triangle. - _Daniel T. Martin_, Apr 13 2022
%C a(n-1) 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
%D Robert B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D Henry Ernest Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
%D Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
%D Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
%D Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
%D 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: Holden-Day, Inc., 1964).
%H T. D. Noe, <a href="/A000124/b000124.txt">Table of n, a(n) for n = 0..1000</a>
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%H Jean-Luc Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%H Jean-Luc Baril and Céline Moreira Dos Santos, <a href="http://jl.baril.u-bourgogne.fr/pancake.pdf">Pizza-cutter's problem and Hamiltonian path</a>, Mathematics Magazine (2019) Vol. 88, No. 1, 1-9.
%H Jean-Luc Baril, Sergey Kirgizov and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1803.06706">Descent distribution on Catalan words avoiding a pattern of length at most three</a>, arXiv:1803.06706 [math.CO], 2018.
%H Jean-Luc Baril, Toufik Mansour and Armen Petrossian, <a href="http://jl.baril.u-bourgogne.fr/equival.pdf">Equivalence classes of permutations modulo excedances</a>, preprint, Journal of Combinatorics, Volume 5 (2014) Number 4.
%H Jean-Luc Baril and Armen Petrossian, <a href="http://jl.baril.u-bourgogne.fr/equivaldescentrevue.pdf">Equivalence classes of permutations modulo descents and left-to-right maxima</a>, preprint, Pure Mathematics and Applications, Volume 25, Issue 1 (Sep 2015).
%H Andrew M. Baxter and Lara K. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf">Ascent sequences avoiding pairs of patterns</a>, preprint, The Electronic Journal of Combinatorics, Volume 22, Issue 1 (2015) Paper #P1.58.
%H Christian Bean, Anders Claesson and Henning Ulfarsson, <a href="http://arxiv.org/abs/1512.03226">Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3</a>, arXiv preprint arXiv:1512.03226 [math.CO], 2017.
%H Henry Bottomley, <a href="/A000124/a000124.gif">Illustration of initial terms</a>
%H Alexander Burstein and Toufik Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, arXiv:math/0112281 [math.CO], 2001.
%H Alexander Burstein and Toufik Mansour, <a href="http://dx.doi.org/10.1007/s000260300000">Words restricted by 3-letter generalized multipermutation patterns</a>, Annals. Combin., 7 (2003), 1-14.
%H Peter M. Chema, <a href="/A000124/a000124_1.pdf">Illustration of first 22 terms as corners of a double square spiral with digital root.</a>
%H David Coles, <a href="http://davcoles.tripod.com">Triangle Puzzle</a>.
%H M. L. Cornelius, <a href="/A006261/a006261_1.pdf">Variations on a geometric progression</a>, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
%H Tom Crawford, <a href="https://www.youtube.com/watch?v=CrvXpnYeKaw">22 Slices of Pizza with Six Cuts</a>, Tom Rocks Maths video (2022)
%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%H Karl Dilcher and Kenneth B. Stolarsky, <a href="http://dx.doi.org/10.1007/s11139-014-9620-5">Nonlinear recurrences related to Chebyshev polynomials</a>, The Ramanujan Journal, 2014, Online Oct. 2014, pp. 1-23. See Cor. 5.
%H Igor Dolinka, James East and Robert D. Gray, <a href="https://doi.org/10.1016/j.jalgebra.2016.09.018">Motzkin monoids and partial Brauer monoids</a>, Journal of Algebra, volume 471, February 2017, pages 251-298. Also preprint <a href="https://arxiv.org/abs/1512.02279">arXiv:1512.02279</a> [math.GR], 2015. See Table 5.
%H Matthew England, Russell Bradford and James H. Davenport, <a href="https://doi.org/10.1016/j.jsc.2019.07.019">Cylindrical algebraic decomposition with equational constraints</a>, Journal of Symbolic Computation, Vol. 100 (2020), pp. 38-71; <a href="https://arxiv.org/abs/1903.08999">arXiv preprint</a>, arXiv:1903.08999 [cs.SC], 2019.
%H J. B. Gil and J. Tomasko, <a href="https://doi.org/10.54550/ECA2022V2S4PP6">Restricted Grassmannian permutations</a>, ECA 2:4 (2022) Article S4PP6.
%H Sahir Gill, <a href="https://doi.org/10.12988/ijma.2018.8537">Bounds for Region Containing All Zeros of a Complex Polynomial</a>, International Journal of Mathematical Analysis (2018), Vol. 12, No. 7, 325-333.
%H Richard K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H M. F. Hasler, <a href="/A000124/a000124.html">Interactive illustration of A000124</a>. [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).]
%H Phillip Tomas Heikoop, <a href="https://digitalcommons.wpi.edu/mqp-all/6822">Dimensions of Matrix Subalgebras</a>, Bachelor's Thesis, Worcester Polytechnic Institute, Massachusetts, 2019.
%H Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%H Cheyne Homberger and Vincent Vatter, <a href="https://doi.org/10.1016/j.jsc.2015.11.019">On the effective and automatic enumeration of polynomial permutation classes</a>, Journal of Symbolic Computation, Vol. 76 (2016), pp. 84-96; <a href="http://arxiv.org/abs/1308.4946">arXiv preprint</a>, arXiv:1308.4946 [math.CO], 2013-2015.
%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n25">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=386">Encyclopedia of Combinatorial Structures 386</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H Myrto Kallipoliti, Robin Sulzgruber, and Eleni Tzanaki, <a href="https://arxiv.org/abs/2006.06949">Patterns in Shi tableaux and Dyck paths</a>, arXiv:2006.06949 [math.CO], 2020.
%H Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H Thomas Langley, Jeffrey Liese and Jeffrey Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order </a>, J. Int. Seq. 14 (2011) # 11.4.2.
%H Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016-2017.
%H Derek Levin, Lara Pudwell, Manda Riehl and Andrew Sandberg, <a href="http://faculty.valpo.edu/lpudwell/slides/georgetown_heaps.pdf">Pattern Avoidance on k-ary Heaps</a>, Slides of Talk, 2014.
%H D. A. Lind, <a href="http://www.fq.math.ca/Scanned/3-4/lind.pdf">On a class of nonlinear binomial sums</a>, Fib. Quart., 3 (1965), 292-298.
%H Jim Loy, <a href="http://web.archive.org/web/20121014155338/http://jimloy.com/puzz/cole.htm">Triangle Puzzle</a>
%H Toufik Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations avoiding a set of patterns from S_3 and a pattern from S_4</a>, arXiv:math/9909019 [math.CO], 1999.
%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016-2018.
%H Johannes W. Meijer and Manuel Nepveu, <a href="https://www.ucbcba.edu.bo/wp-content/uploads/PDF/Acta-Nova/v4.n1.Meijer.pdf">Euler's ship on the Pentagonal Sea</a>, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.
%H Markus Moll, <a href="https://arxiv.org/abs/1312.5136">On a family of random noble means substitutions</a>, Dr. Math. Dissertation, Universität Bielefeld, 2013, arXiv:1312.5136 [math.DS], 2013.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Derek J. Price, <a href="http://www.jstor.org/stable/3609091">Some unusual series occurring in n-dimensional geometry</a>, Math. Gaz., Vol. 30, No. 290 (1946), pp. 149-150.
%H Lara Pudwell and Andrew Baxter, <a href="http://faculty.valpo.edu/lpudwell/slides/pp2014_pudwell.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1712.06543">Enumerating the states of the twist knot</a>, arXiv:1712.06543 [math.CO], 2017.
%H Franck Ramaharo and Fanja Rakotondrajao, <a href="https://arxiv.org/abs/1712.04026">A state enumeration of the foil knot</a>, arXiv:1712.04026 [math.CO], 2017.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%H Nathan Reading, <a href="http://www4.ncsu.edu/~nreadin/papers/thesis.pdf">On the structure of Bruhat Order</a>, Ph.D. dissertation, University of Minnesota, April 2002.
%H Nathan Reading, <a href="http://www4.ncsu.edu/~nreadin/papers/dissective.pdf">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets</a>
%H Nathan Reading, <a href="http://doi.org/10.1023/A:1015287106470">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets</a>, Order, Vol. 19, no. 1 (2002), 73-100.
%H Herman P. Robinson, <a href="/A000124/a000124.pdf">Letter to N. J. A. Sloane, Aug 16 1971, with attachments</a>
%H Rodica Simion and Frank W. Schmidt, <a href="http://dx.doi.org/10.1016/S0195-6698(85)80052-4">Restricted permutations</a>, European J. Combin., 6, 383-406, 1985; see Example 3.5.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/dijen.pdf">On single-deletion-correcting codes</a>, 2002.
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 1.
%H Andrew James Turner and Julian Francis Miller, <a href="http://andrewjamesturner.co.uk/files/YDS2014.pdf">Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences</a>, 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircleDivisionbyLines.html">Circle Division by Lines</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlaneDivisionbyLines.html">Plane Division by Lines</a>.
%H Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http://www.math.nthu.edu.tw/~amen/2008/2008.htm">Applied Mathematics Electronic Notes</a>, Vol. 8 (2008), pp. 45-52.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Floyd%27s_triangle">Floyd's triangle</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1 - x + x^2)/(1 - x)^3. _Simon Plouffe_ in his 1992 dissertation
%F a(n) = A108561(n+3, 2). - _Reinhard Zumkeller_, Jun 10 2005
%F 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
%F Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - _Michael Somos_, Sep 04 2006
%F 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
%F a(n) = A000217(n) + 1.
%F a(n) = a(n-1) + n. E.g.f.:(1 + x + x^2/2)*exp(x). - _Geoffrey Critzer_, Mar 10 2009
%F a(n) = Sum_{k = 0..n + 1} binomial(n+1, 2(k - n)). - _Paul Barry_, Aug 29 2004
%F a(n) = binomial(n+2, 1) - 2*binomial(n+1, 1) + binomial(n+2, 2). - _Zerinvary Lajos_, May 12 2006
%F From _Thomas Wieder_, Feb 25 2009: (Start)
%F 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)
%F a(n) = A034856(n+1) - A005843(n) = A000217(n) + A005408(n) - A005843(n). - _Jaroslav Krizek_, Sep 05 2009
%F a(n) = 2*a(n-1) - a(n-2) + 1. - _Eric Werley_, Jun 27 2011
%F 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
%F a(n) = A014132(n, 1) for n > 0. - _Reinhard Zumkeller_, Dec 12 2012
%F a(n) = 1 + floor(n/2) + ceiling(n^2/2) = 1 + A004526(n) + A000982(n). - _Wesley Ivan Hurt_, Jun 14 2013
%F a(n) = A228074(n+1, n). - _Reinhard Zumkeller_, Aug 15 2013
%F For n > 0: A228446(a(n)) = 3. - _Reinhard Zumkeller_, Mar 12 2014
%F a(n) >= A263883(n) and a(n(n-1)/2) >= A055503(n). - _Jonathan Sondow_, Dec 01 2015
%F From _Ilya Gutkovskiy_, Jun 29 2016: (Start)
%F Dirichlet g.f.: (zeta(s-2) + zeta(s-1) + 2*zeta(s))/2.
%F Sum_{n >= 0} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = A226985. (End)
%F a(n) = (n+1)^2 - A000096(n). - _Anton Zakharov_, Jun 29 2016
%F a(n) = A101321(1, n). - _R. J. Mathar_, Jul 28 2016
%F a(n) = 2*a(n-1) - binomial(n-1, 2) and a(0) = 1. - _Armend Shabani_, Mar 10 2017
%F a(n) = A002620(n+2) + A002620(n-1). - _Anton Zakharov_, May 11 2017
%F From _Klaus Purath_, Jan 29 2020: (Start)
%F a(n) = (Sum_{i=n-2..n+2} A000217(i))/5.
%F a(n) = (Sum_{i=n-2..n+2} A002378(i))/10.
%F a(n) = (Sum_{i=n..n+2} A002061(i)+1)/6.
%F a(n) = (Sum_{i=n-1..n+2} A000290(i)+2)/8.
%F a(n) = A060533(n-1) + 10, n > 5.
%F a(n) = (A002378(n) + 2)/2.
%F a(n) = A152948(n+2) - 1.
%F a(n) = A152950(n+1) - 2.
%F a(n) = (A002061(n) + A002061(n+2))/4.
%F (End)
%F Sum_{n>=0} (-1)^n/a(n) = A228918. - _Amiram Eldar_, Nov 20 2020
%F From _Amiram Eldar_, Feb 17 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(15)*Pi/2)*sech(sqrt(7)*Pi/2).
%F Product_{n>=1} (1 - 1/a(n)) = 2*Pi*sech(sqrt(7)*Pi/2). (End)
%F a((n^2-3n+6)/2) + a((n^2-n+4)/2) = a(n^2-2n+6)/2. - _Charlie Marion_, Feb 14 2023
%e a(3) = 7 because the 132- and 321-avoiding permutations of {1, 2, 3, 4} are 1234, 2134, 3124, 2314, 4123, 3412, 2341.
%e G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 22*x^6 + 29*x^7 + ...
%p A000124 := n-> n*(n+1)/2+1;
%t FoldList[#1 + #2 &, 1, Range@ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Accumulate[Range[0, 60]] + 1 (* _Harvey P. Dale_, Mar 12 2013 *)
%t Select[Range[2000], IntegerQ[Sqrt[8 # - 7]] &] (* _Vincenzo Librandi_, Apr 16 2014 *)
%t Table[PolygonalNumber[n] + 1, {n, 0, 52}] (* _Michael De Vlieger_, Jun 30 2016, Version 10.4 *)
%t LinearRecurrence[{3, -3, 1}, {1, 2, 4}, 53] (* _Jean-François Alcover_, Sep 23 2017 *)
%o (PARI) {a(n) = (n^2 + n) / 2 + 1}; /* _Michael Somos_, Sep 04 2006 */
%o (Haskell)
%o a000124 = (+ 1) . a000217
%o -- _Reinhard Zumkeller_, Oct 04 2012, Nov 15 2011
%o (Magma) [n: n in [0..1500] | IsSquare(8*n-7)]; // _Vincenzo Librandi_, Apr 16 2014
%o (GAP) List([0..60],n->n*(n+1)/2+1); # _Muniru A Asiru_, Apr 11 2018
%o (Scala) (1 to 52).scanLeft(1)(_ + _) // _Alonso del Arte_, Feb 24 2019
%o (Python)
%o def a(n): return n*(n+1)//2 + 1
%o print([a(n) for n in range(53)]) # _Michael S. Branicky_, Aug 26 2021
%Y Cf. A000096 (Maximal number of pieces that can be obtained by cutting an annulus with n cuts, for n >= 1).
%Y Slicing a cake: A000125, a bagel: A003600.
%Y Partial sums =(A033547)/2, (A014206)/2.
%Y The first 20 terms are also found in A025732 and A025739.
%Y 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.
%Y Cf. A055469 Quasi-triangular primes.
%Y Cf. A002620.
%Y Cf. A000217
%K nonn,core,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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