A006522 -id:A006522 - cp's OEIS frontend

A007678 Number of regions in regular n-gon with all diagonals drawn.

0, 0, 1, 4, 11, 24, 50, 80, 154, 220, 375, 444, 781, 952, 1456, 1696, 2500, 2466, 4029, 4500, 6175, 6820, 9086, 9024, 12926, 13988, 17875, 19180, 24129, 21480, 31900, 33856, 41416, 43792, 52921, 52956, 66675, 69996, 82954, 86800, 102050, 97734, 124271, 129404, 149941

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

This sequence and A006533 are two equivalent ways of presenting the same sequence.
A quasipolynomial of order 2520. - Charles R Greathouse IV, Jan 15 2013
Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
This sequence only counts polygons, in contrast to A006533 which also counts the n segments of the circumscribed circle delimited by the edges of the regular n-gon. Therefore a(n) = A006533(n) - n. See also A006561 which counts the intersection points, and A350000 which considers iterated "cutting along diagonals". - M. F. Hasler, Dec 13 2021
The Petrie polygon orthographic projection of a regular n-simplex is a regular (n+1)-gon with all diagonals drawn. Hence a(n+1) is the number of regions in the Petrie polygon of a regular n-simplex. - Mohammed Yaseen, Nov 05 2022

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Robert Dougherty-Bliss, First draft of Python program to produce colored drawings of these figures, Github, Feb 09 2020.
Martin Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons (in German).
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [Author's copy]. The latest arXiv version arXiv:math/9508209 has corrected some typos in the published version.
Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
M. Rubinstein, Drawings for n=4,5,6,...
Scott R. Shannon, Colored illustration for n = 17
Scott R. Shannon, Colored illustration for n = 18
Scott R. Shannon, Colored illustration for n = 19
Scott R. Shannon, Colored illustration for n = 23
Scott R. Shannon, Colored illustration for n = 27
Scott R. Shannon, Colored illustration for n = 40
Scott R. Shannon, Colored illustration for n = 41 (1st version)
Scott R. Shannon, Colored illustration for n = 41 (2nd version)
Scott R. Shannon, Colored illustration for n = 41 (3rd version). This variation has coloring based on the number of edges of the polygon: red = 3-gon, orange = 4-gon, yellow = 5-gon, light-green = 6-gon etc.
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18.
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals
Sequences formed by drawing all diagonals in regular polygon
Sequences related to chord diagrams

Cf. A001006, A054726, A006533, A006561, A006600, A007569 (number of vertices), A006522, A135565 (number of line segments).
A062361 gives number of triangles, A331450 and A331451 give distribution of polygons by number of sides.
A333654, A335614, A335646, A337330 give the number of internal n-gon to k-gon contacts for n>=3, k>=n.
A187781 gives number of distinct regions.

del[m_,n_]:=If[Mod[n,m]==0,1,0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n]; Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

/* Only for odd n > 3, not suitable for other values of n! */ { a(n)=local(nr,x,fn,cn,fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0,nr+=x*(x+1)*(2*x+1)/6*n); nr; } \\ Jon Perry, Jul 08 2003

apply( {A007678(n)=if(n%2, (((n-6)*n+23)*n-42)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 31, 40) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A007678(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 84*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

For odd n > 3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n-5)/2. Simplifying the floor/ceiling components gives the PARI code below. - Jon Perry, Jul 08 2003
For odd n, a(n) = (24 - 42*n + 23*n^2 - 6*n^3 + n^4)/24. - Graeme McRae, Dec 24 2004
a(n) = A006533(n) - n. - T. D. Noe, Dec 23 2006
For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...]. - Gary W. Adamson, Aug 02 2011
a(n) = A135565(n) - A007569(n) + 1. - Max Alekseyev
See the Mma code in A006533 for the explicit Poonen-Rubenstein formula that holds for all n. - N. J. A. Sloane, Jan 23 2020

More terms from Graeme McRae, Dec 26 2004

a(1) = a(2) = 0 prepended by Max Alekseyev, Dec 01 2011

A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500, 94163002754652, 369507926510352

Offset: 1

Barry E. Williams, Dec 19 1999

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen, Dec 10 2001
From Benoit Cloitre, Jan 29 2002: (Start)
Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
  1  2  3  4 ...
  2  4  7 11 ...
  3  7 14 25 ...
  4 11 25 50 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, Jan 31 2005
If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - Gary W. Adamson, May 15 2009
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013
From R. J. Mathar, Jun 30 2013: (Start)
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
  1,   1,   1,   1,   1,   1,
  1,   2,   3,   4,   5,   6,
  1,   3,   6,  10,  15,  21,
  1,   4,  10,  20,  35,  56,
  1,   5,  15,  35,  70, 126,
  1,   6,  21,  56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
From Gus Wiseman, Apr 12 2019: (Start)
Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
  (1)  (2)   (3)
       (11)  (31)
       (21)  (32)
       (22)  (33)
             (111)
             (211)
             (221)
             (222)
             (311)
             (321)
             (322)
             (331)
             (332)
             (333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - Lee A. Newberg, Jul 20 2024

			Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, arXiv:math/0611106 [math.CO], 2006-2007.
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 10.
F. Chapoton, Clusters.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
Hugh Thomas, Tamari Lattices and Non-Crossing Partitions in Types B and D, arXiv:math/0311334 [math.CO], 2003-2005.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study, Vol. 56, No. 1 (2023), 1-17.

Left-central elements of the (1, 2)-Pascal triangle A029635.
Column sums of A096771.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064; first differences of A000984.
Cf. A097613.
Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.

a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
-- Reinhard Zumkeller, May 25 2013

[Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016

C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # Emeric Deutsch, Jan 08 2008
Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # Zerinvary Lajos, Jan 01 2007
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015

Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* Harvey P. Dale, Jan 15 2012 *)

a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ Charles R Greathouse IV, Jun 25 2013

{a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 08 2014

{a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014
G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - Paul D. Hanna, Nov 08 2014
a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - Jon Perry Apr 13 2004
a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - Gary W. Adamson, May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos, Dec 20 2005
a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006
The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - Paul Barry, Oct 17 2009
a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012
D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - Alois P. Heinz, Apr 25 2014
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
a(n) = 2 * A097613(n) for n > 1. - Bruce J. Nicholson, Jan 06 2019
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar

A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 9, 5, 1, 15, 16, 12, 6, 1, 21, 25, 22, 15, 7, 1, 28, 36, 35, 28, 18, 8, 1, 36, 49, 51, 45, 34, 21, 9, 1, 45, 64, 70, 66, 55, 40, 24, 10, 1, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 78, 121, 145, 153, 148, 133, 111

Offset: 1

Clark Kimberling, Jul 14 2003

The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
This is the accumulation array (cf. A144112) of A144257 (which is the weight array of this sequence). - Clark Kimberling, Sep 16 2008
By rows, the sequence beginning (1, N, ...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0, ...); and is the second partial sum of (1, (N-2), (N-2), (N-2), ...). Example: The sequence (1, 4, 9, 16, 25, ...) is the binomial transform of (1, 3, 2, 0, 0, 0, ...) and the second partial sum of (1, 2, 2, 2, ...). - Gary W. Adamson, Aug 23 2015

			First 6 rows:
=========================================
n\k|  1   2    3    4    5    6     7
---|-------------------------------------
1  |  1   3    6   10   15   21    28 ... (A000217, triangular numbers)
2  |  1   4    9   16   25   36    49 ... (A000290, squares)
3  |  1   5   12   22   35   51    70 ... (A000326, pentagonal numbers)
4  |  1   6   15   28   45   66    91 ... (A000384, hexagonal numbers)
5  |  1   7   18   34   55   81   112 ... (A000566, heptagonal numbers)
6  |  1   8   21   40   65   96   133 ... (A000567, octagonal numbers)
...
The array formed by the complements: A183225.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 76 at p. 189.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Wikipedia, Polygonal number: Table of values.

Cf. A006522, A057145, A086271, A086272, A086273, A139601, A183225.
Cf. A000217, A000290, A000326, A000384, A000566, A000567.
Cf. A114112, A144257 .

T:=func; [T(k,n-k+1): k in [1..n], n in [1..12]]; // Bruno Berselli, Dec 19 2014

t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten

T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).
2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). - Bruno Berselli, Dec 19 2014
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)

Extended by Clark Kimberling, Jan 01 2011

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

Original entry on oeis.org

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

Offset: 0

Bruno Berselli, Jan 31 2014

After 0, first trisection of A011779 and right border of A177708.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

[n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 12, 16, 15, 1, 7, 15, 22, 25, 21, 1, 8, 18, 28, 35, 36, 28, 1, 9, 21, 34, 45, 51, 49, 36, 1, 10, 24, 40, 55, 66, 70, 64, 45, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66, 1, 13, 33, 58, 85, 111, 133, 148, 153, 145, 121, 78

Offset: 1

Clark Kimberling, Jul 14 2003

The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

			Columns 1,2,3 are the triangular, square and pentagonal numbers.
Northwest corner:
       k=1 k=2 k=3 k=4 k=5
  n=1:   1   1   1   1   1 ...
  n=2:   3   4   5   6   7 ...
  n=3:   6   9  12  15  18 ...
  n=4:  10  16  22  28  34 ...
  n=5:  15  25  35  45  55 ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 diagonals, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A006522, A086270, A086272, A086273.

Main diagonal gives A006000(n-1).

T[n_, k_] := PolygonalNumber[k+2, n]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 04 2016 *)

T(n, k) = k*C(n,2) + n.
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).
G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)

A059173 Maximal number of regions into which 4-space can be divided by n hyperspheres.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 62, 114, 198, 326, 512, 772, 1124, 1588, 2186, 2942, 3882, 5034, 6428, 8096, 10072, 12392, 15094, 18218, 21806, 25902, 30552, 35804, 41708, 48316, 55682, 63862, 72914, 82898, 93876, 105912, 119072, 133424, 149038

Offset: 0

N. J. A. Sloane, Feb 15 2001

n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i=0..k} C(n, i) regions.
From Raphie Frank Nov 24 2012: (Start)
Define the gross polygonal sum, GPS(n), of an n-gon as the maximal number of combined points (p), intersections (i), connections (c = edges (e) + diagonals (d)) and areas (a) of a fully connected n-gon, plus the area outside the n-gon. The gross polygonal sum (p + i + c + a + 1) is equal to this sequence and, for all n > 0, then individual components of this sum can be calculated from the first 5 entries in row (n-1) of Pascal's triangle.
For example, the gross polygonal sum of a 7-gon (the heptagon):
Let row 6 of Pascal's triangle = {1, 6, 15, 20, 15, 6, 1} = A B C D E F G.
Points = 1 + 6 = A + B = 7 [A000027(n)].
Intersections = 20 + 15 = D + E = 35 [A000332(n+2)].
Connections = 6 + 15 = B + C = 21 [A000217(n)].
Areas inside = 15 + 20 + 15 = C + D + E = 50 [A006522(n+1)].
Areas outside = 1 = A = 1 [A000012(n)].
Then, GPS(7) = 7 + 35 + 21 + 50 + 1 = 2(A + B + C + D + E) = 114 = a(7). In general, a(n) = GPS(n). (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
A row of A059250.
Cf. A000012, A000027, A000217, A000332, A006522.

LinearRecurrence[{5,-10,10,-5,1},2^Range[0,5],50] (* Paolo Xausa, Dec 29 2023 *)

a(0) = 1; a(n) = 2 * A000127(n), for n >= 1.
G.f.: -(x^5 + x^4 - 2*x^3 + 4*x^2 - 3*x + 1)/(x-1)^5. - Colin Barker, Oct 06 2012
E.g.f.: exp(x)*(2 + x^2 + x^4/12) - 1. - Stefano Spezia, May 19 2024

A027927 Number of plane regions after drawing (in general position) a convex n-gon and all its diagonals.

Original entry on oeis.org

1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176, 7526, 9087, 10880, 12927, 15251, 17876, 20827, 24130, 27812, 31901, 36426, 41417, 46905, 52922, 59501, 66676, 74482, 82955, 92132, 102051, 112751, 124272, 136655, 149942, 164176, 179401

Offset: 2

Clark Kimberling

For n>=1, a(n+1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 5 with exactly one descent. - Jessica A. Tomasko, Nov 15 2022

			a(2)=1 (segment traced twice has only exterior).

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
Lapo Cioni and Luca Ferrari, Enumerative Results on the Schröder Pattern Poset, In: Dennunzio A., Formenti E., Manzoni L., Porreca A. (eds) Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, Lecture Notes in Computer Science, vol 10248.
Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013
J. B. Gil and J. Tomasko, Restricted Grassmannian permutations, ECA 2:4 (2022) Article S4PP6.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006522 (does not count exterior of n-gon).

Cf. A000045, A000217, A004006, A027926, A228074.

List([2..50], n-> (n^4 -6*n^3 +23*n^2 -42*n +48)/24); # G. C. Greubel, Sep 06 2019

[(n^4 -6*n^3 +23*n^2 -42*n +48)/24: n in [2..50]]; // G. C. Greubel, Sep 06 2019

seq((n^4 -6*n^3 +23*n^2 -42*n +48)/24, n=2..50); # G. C. Greubel, Sep 06 2019

LinearRecurrence[{5,-10,10,-5,1 }, {1,2,5,12,26}, 50] (* Vincenzo Librandi, Feb 01 2012 *)
S[n_] :=n*(n+1)/2; Table[S[S[n]+2]/3, {n, 0, 50}] (* Waldemar Puszkarz, Jan 22 2016 *)

a(n)=n*(n^3-6*n^2+23*n-42)/24+2 \\ Charles R Greathouse IV, Jan 31 2012

[(n^4 -6*n^3 +23*n^2 -42*n +48)/24 for n in (2..50)] # G. C. Greubel, Sep 06 2019

a(n) = T(n, 2*n-4), T given by A027926.
a(n) = 1 + binomial(n, 4) + binomial(n-1, 2) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 48)/24.
G.f.: x^2*(1 -3*x +5*x^2 -3*x^3 +x^4)/(1-x)^5. - Colin Barker, Jan 31 2012
a(n) = (1/6)*A152950(n-1)*A152948(n). - Bruno Berselli, Jan 31 2012
a(n) = A000217(A000217(n-2)+2)/3, a(n+1) - a(n) = A004006(n-1) for n > 2. - Waldemar Puszkarz, Jan 22 2016 [Adjusted for offset by Peter Munn, Jan 10 2023]
a(n) = 1 + Sum {i=3..5} binomial(n-1, i-1). - Jessica A. Tomasko, Nov 15 2022

New name from Len Smiley, Oct 19 2001

A185507 Second accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 4, 5, 11, 19, 15, 25, 49, 55, 35, 50, 105, 136, 125, 70, 91, 200, 280, 300, 245, 126, 154, 350, 515, 600, 575, 434, 210, 246, 574, 875, 1075, 1125, 1001, 714, 330, 375, 894, 1400, 1785, 1975, 1925, 1624, 1110, 495, 550, 1335, 2136, 2800, 3220, 3325, 3080, 2496, 1650, 715, 781, 1925, 3135, 4200, 4970, 5341, 5250, 4680, 3675, 2365, 1001, 1079, 2695, 4455, 6075, 7350, 8134, 8330, 7890, 6825, 5225, 3289, 1365, 1456, 3679, 6160, 8525, 10500, 11886, 12544, 12390, 11400, 9625, 7216, 4459, 1820, 1925, 4914, 8320, 11660

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of accumulation array (aa).

Sequence is aa(aa(A000027)).

			Northwest corner:
   1,   4,  11,   25,   50,   91,  154
   5,  19,  49,  105,  200,  350,  574
  15,  55, 136,  280,  515,  875, 1400
  35, 125, 300,  600, 1075, 1785, 2800
  70, 245, 575, 1125, 1975, 3220, 4970

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A006522 (row 1), A000332 (column 1).

Cf. A000027, A185506, A185508, A185509.

g[n_,k_]:=k*n(k+1)(n+1)(3n^2+(4k+11)n+3k^2-k+16)/144;
TableForm[Table[g[n,k],{n,1,10},{k,1,15}]]
Table[g[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*n*(k+1)*(n+1)*(3*n^2 + (4*k+11)*n + 3*k^2 - k + 16)/144.

A008498 4-dimensional centered tetrahedral numbers.

Original entry on oeis.org

1, 6, 21, 56, 126, 251, 456, 771, 1231, 1876, 2751, 3906, 5396, 7281, 9626, 12501, 15981, 20146, 25081, 30876, 37626, 45431, 54396, 64631, 76251, 89376, 104131, 120646, 139056, 159501, 182126

Offset: 0

N. J. A. Sloane

Binomial transform of (1,5,10,10,5,0,0,0,...). - Paul Barry, Jul 01 2003
If X is an n-set and Y a fixed 5-subset of X then a(n-5) is equal to the number of 5-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Also the sum of five consecutive terms of A000332. - Bruno Berselli, Jun 18 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 224 (general formula for n-th centered polytope number).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A005894, A006522.

List([0..40], n-> Binomial(n+5,5) - Binomial(n,5)); # G. C. Greubel, Nov 08 2019

[(5*n^4+10*n^3+55*n^2+50*n+24)/24: n in [0..30] ]; // Vincenzo Librandi, Aug 21 2011

[Binomial(n+5,5) - Binomial(n,5): n in [0..40]]; // G. C. Greubel, Nov 08 2019

[seq(binomial(n+5,5)-binomial(n,5), n=0..45)]; # Zerinvary Lajos, Jul 21 2006

LinearRecurrence[{5,-10,10,-5,1}, {1,6,21,56,126}, 40] (* Harvey P. Dale, Dec 18 2013 *)
Table[1 + 5n(n+1)(n^2 +n +10)/24, {n, 0, 40}] (* Bruno Berselli, Jun 18 2015 *)

[binomial(n+5,5) - binomial(n,5) for n in (0..40)] # G. C. Greubel, Nov 08 2019

G.f.: (1-x^5)/(1-x)^6 = (1 +x +x^2 +x^3 +x^4)/(1-x)^5.
a(n) = C(n,0) + 5*C(n,1) + 10*C(n,2) + 10*C(n,3) + 5*C(n,4). - Paul Barry, Jul 01 2003
a(n) = (5*n^4 + 10*n^3 + 55*n^2 + 50*n + 24)/24. - Paul Barry, Jul 01 2003
a(n) = binomial(n+5,5) - binomial(n,5). - Zerinvary Lajos, Jul 21 2006
a(n) = 1 + 5*A006522(n+2). - Bruno Berselli, Jun 18 2015
E.g.f.: (24 + 120*x + 120*x^2 + 40*x^3 + 5*x^4)*exp(x)/24. - G. C. Greubel, Nov 08 2019

cp's OEIS Frontend

A007678 Number of regions in regular n-gon with all diagonals drawn.

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A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

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A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A236770 a(n) = n*(n + 1)*(3*n^2 + 3*n - 2)/8.

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A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

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A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.