A006522 -id:A006522 - cp's OEIS frontend

A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

Original entry on oeis.org

1, 4, 11, 24, 47, 80

Offset: 3

Vladimir Letsko, May 29 2009, May 30 2009, Apr 20 2010

It seems that a(9)=137 and a(n) = A007678(n) for all even n.

Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8
Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8 [Cached copy, pdf version only]
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian)
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian). [Cached copy, pdf version only]
V. A. Letsko, M. A. Voronina, Classification of convex polygons, Grani Poznaniya, 1(11), 2011. (in Russian)
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon.

Cf. A006522, A007678, A230281.

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.

Original entry on oeis.org

0, 1, 14, 61, 175, 400, 791, 1414, 2346, 3675, 5500, 7931, 11089, 15106, 20125, 26300, 33796, 42789, 53466, 66025, 80675, 97636, 117139, 139426, 164750, 193375, 225576, 261639, 301861, 346550, 396025, 450616, 510664, 576521, 648550, 727125, 812631, 905464, 1006031

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of centered 11-gonal (or hendecagonal) pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004467.

Cf. similar sequences provided by the partial sums of centered k-gonal pyramidal numbers: A006522 (k=1), A006007 (k=2), A002817 (k=3), A006325 (k=4), A006322 (k=5), A000537 (k=6), A006323 (k=7), A006324 (k=8), A236770 (k=9), A264853 (k=10), this sequence (k=11), A062392 (k=12), A264888 (k=13).

[n*(n+1)*(11*n^2+11*n-10)/24: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (11 n^2 + 11 n - 10)/24, {n, 0, 50}]

a(n)=n*(n+1)*(11*n^2+11*n-10)/24 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 9*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004467(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

Original entry on oeis.org

1, 4, 11, 25, 50, 91, 154, 234, 375, 550, 769, 1079, 1456, 1783, 2500, 3196, 3987, 5016, 6175, 7348, 9086, 10879, 12836, 15250, 17875, 20682, 24129, 27811, 31419, 36425, 41416, 46664, 52921, 59500, 66489, 74481, 82954, 91807, 102050, 112750, 123700, 136654

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

nonn

A semicircular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.

Lars Blomberg, Table of n, a(n) for n = 1..80
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 7.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 17.
Scott R. Shannon, Illustration for n = 19.
Scott R. Shannon, Illustration for n = 20.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
Wikipedia, Semicircle.

Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

More terms from Lars Blomberg, Apr 20 2020

A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

Original entry on oeis.org

3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560

Offset: 3

Mohammad K. Azarian

There is a connection to A014626: number of intersection points of diagonals of n-gon, plus number of vertices, b(n) = n*(n+1)*(n^2-7*n+18)/24 and A006522: number of regions created by sides and diagonals of n-gon, c(n) = (n-1)*(n-2)*(n^2-3*n+12)/24. These are related by the Euler-formula: b(n) + c(n) - a(n) = 1. - Georg Wengler, Mar 31 2005

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A014626, A006522, A134393.

Table[Binomial[n,2]+2Binomial[n,4],{n,3,50}] (* Harvey P. Dale, Oct 03 2020 *)

a(n) = (n^4-6*n^3+17*n^2-24*n)/12 + n; or equally n*(n-1)*(n^2-5*n+12)/12.
G.f.: x^3*(3-7*x+10*x^2-5*x^3+x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = C(n,2) + 2*C(n,4). - Gary Detlefs, Jun 06 2010

G.f. proposed by Maksym Voznyy, checked and corrected by R. J. Mathar, Sep 16 2009
More terms from Erich Friedman
Offset corrected by Mohammad K. Azarian, Nov 19 2008
Offset corrected by Eric Rowland, Aug 15 2017

A125230 Triangle T(n,k) (0<=k<=n) read by rows in which column k contains the binomial transform of the sequence of k 0's, (k+1) 1's, followed by 0's.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 11, 5, 1, 1, 15, 25, 16, 6, 1, 1, 21, 50, 42, 22, 7, 1, 1, 28, 91, 98, 64, 29, 8, 1, 1, 36, 154, 210, 163, 93, 37, 9, 1, 1, 45, 246, 420, 381, 256, 130, 46, 10, 1, 1, 55, 375, 792, 837, 638, 386, 176, 56, 11, 1, 1, 66, 550, 1419, 1749, 1485, 1024

Offset: 0

Gary W. Adamson, Nov 24 2006

A125231 is another triangle with the same row sums A045623: (1, 2, 5, 12, 28, 64, 144, 320...).

			T(5,2) = C(5,2) + C(5,3) + C(5,4) = 10 + 10 + 5 = 25.
First few rows of the triangle are:
1
1 1
1 3 1
1 6 4 1
1 10 11 5 1
1 15 25 16 6 1

Cf. A007318, A125231. Columns k=0-3 give: A000012, A000217, A006522(n+1), A055796(n-3). Row sums give: A045623.

T:= (n, k)-> add (binomial (n, j), j=k..min(2*k, n)): seq (seq (T(n, k), k=0..n), n=0..12);

T(n,k) = Sum_{j=k..min(2*k,n)} C(n,j).

Edited with more terms and Maple program by Alois P. Heinz, Oct 16 2009

A128141 A122432 (unsigned) * A128132.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 8, 4, 5, 14, 15, 11, 5, 6, 20, 24, 21, 14, 6, 7, 27, 35, 34, 27, 17, 7, 8, 35, 48, 50, 44, 33, 20, 8, 9, 44, 63, 69, 65, 54, 39, 23, 9, 10, 54, 80, 91, 90, 80, 64, 45, 26, 10

Offset: 1

Gary W. Adamson, Feb 16 2007

Row sums = A006522 starting (1, 4, 11, 25, 50, 91, ...). A128142 = A128132 * A122432.

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,  3;
  4,  9,  8,  4;
  5, 14, 15, 11,  5;
  6, 20, 24, 21, 14,  6;
  7, 27, 35, 34, 27, 17,  7;
  ...

Cf. A122432, A128132, A006522, A128142, A006325.

Unsigned A122432 * A128132 as infinite lower triangular matrices; where unsigned A122432 = (1; 3, 1; 6, 3, 1; ...).

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Original entry on oeis.org

1, 4, 11, 26, 55, 105, 184, 301, 466, 690, 985, 1364, 1841, 2431, 3150, 4015, 5044, 6256, 7671, 9310, 11195, 13349, 15796, 18561, 21670, 25150, 29029, 33336, 38101, 43355, 49130, 55459, 62376, 69916, 78115, 87010, 96639, 107041, 118256, 130325

Offset: 1

Dan Graybill (clopen(AT)comcast.net), Aug 12 2008

Consider the array of triangular, square and centered polygonal numbers (irregular variant of A086272 and A086273):
 3   6  10  15  21  28  36  45  55  A000217
 4   9  16  25  36  49  64  81 100  A000290
 6  16  31  51  76 106 141 181 226  A005891
 7  19  37  61  91 127 169 217 271  A003215
 8  22  43  71 106 148 197 253 316  A069099
 9  25  49  81 121 169 225 289 361  A016754
10  28  55  91 136 190 253 325 406  A060544
11  31  61 101 151 211 281 361 451  A062786
12  34  67 111 166 232 309 397 496  A069125
13  37  73 121 181 253 337 433 541  A003154
14  40  79 131 196 274 365 469 586  A069126
15  43  85 141 211 295 393 505 631  A069127
etc. The sequence contains the antidiagonal sums of this array. - R. J. Mathar, Jun 05 2011
For comparison, the antidiagonal sums of A086270 are essentially A006522 starting at the 4th term. - R. J. Mathar, Sep 20 2008

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

Cf. A000217.

a(n) = (n-1)*(n^3+11*n^2-38*n+120)/24, n>1. - R. J. Mathar, Sep 12 2008

G.f.: x*(1-x+x^2+x^3-x^5)/(1-x)^5. - Alexander R. Povolotsky, Jun 06 2011

A230150 Irregular triangle read by rows: Possible numbers of pieces resulting from cutting a convex n-sided polygon along all its diagonals.

Original entry on oeis.org

1, 4, 11, 24, 25, 47, 48, 49, 50, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 137

Offset: 3

Vladimir Letsko, Oct 11 2013

Beginning from number of sides equal to 18 the terms no longer increase between rows. For example, the number of pieces for the regular 18-gon is fewer than the number of pieces for regular 17-gon.

Obviously there exists a number k_0 such that k_0 is not in the sequence and k is in the sequence for all k > k_0.

			The beginning of the irregular triangle is:
3| 1
4| 4
5| 11
6| 24, 25
7| 47, 48, 49, 50,
8| 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91
9| 137 (incomplete)

V.A. Letsko, M.A. Voronina Classification of convex polygons, Grani Poznaniya, 1(11), 2011. (in Russian)
Vladimir Letsko, Mathematical Marathon at vspu, Problem 102 (in Russian)
Vladimir Letsko Illustration of all cases for number of sides from 3 to 8

Cf. A006522, A160860, A007678.

a(n,s_1,...,s_m) = A006522(n) - sum_{k=1}^m s_k*k*(k+1)/2, where m = floor(n/2)-2 and s_k denotes number of inner points in which exactly k+2 diagonals are intersected.

A337977 Triangle T(n,m) = C(n-1,n-m)Sum_{k=1..n} C(2k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 8, 5, 1, 10, 22, 26, 14, 1, 15, 50, 85, 90, 42, 1, 21, 100, 225, 348, 322, 132, 1, 28, 182, 525, 1050, 1442, 1176, 429, 1, 36, 308, 1120, 2730, 4928, 5992, 4356, 1430, 1, 45, 492, 2226, 6426, 14238, 22920, 24894, 16302, 4862

Offset: 1

Vladimir Kruchinin, Oct 05 2020

			1,
1, 1,
1, 3,  2,
1, 6,  8,  5,
1,10, 22, 26, 14,
1,15, 50, 85, 90, 42,
1,21,100,225,348,322,132

T(2*n,n) is A069720.

2nd column: A000217, 3rd column: 2*A006522 or 2*(A027927-1).

Table[Binomial[n - 1, n - m] Sum[Binomial[2 k - 2, k - 1] Binomial[n - m, m - k]/m, {k, n}], {n, 10}, {m, n}] // Flatten (* Michael De Vlieger, Oct 05 2020 *)

T(n,m):=(binomial(n-1,n-m)*sum(binomial(2*k-2,k-1)*binomial(n-m,m-k),k,1,n))/m;

G.f.: A(x,y) = -(sqrt((2*sqrt(-4*x^2*y+x^2-2*x+1)+3*x-2)/(4*x))-1/2).

cp's OEIS Frontend

A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

Views

Author

Keywords

Comments

Links

Crossrefs

A264854 a(n) = n*(n + 1)*(11*n^2 + 11*n - 10)/24.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A125230 Triangle T(n,k) (0<=k<=n) read by rows in which column k contains the binomial transform of the sequence of k 0's, (k+1) 1's, followed by 0's.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A128141 A122432 (unsigned) * A128132.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A230150 Irregular triangle read by rows: Possible numbers of pieces resulting from cutting a convex n-sided polygon along all its diagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A337977 Triangle T(n,m) = C(n-1,n-m)*Sum_{k=1..n} C(2*k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

Views

Author

Keywords

Examples

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.

A337977 Triangle T(n,m) = C(n-1,n-m)Sum_{k=1..n} C(2k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.