A255011 -id:A255011 - cp's OEIS frontend

A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

1, 5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, 220113, 286357, 379097, 551669, 606241, 880293, 951445, 1209049, 1507521, 1947877, 2002669, 2624409, 3056093, 3550425, 3955049, 5069037, 5062153, 6669665, 7081969, 8143193, 9365089, 10296469

Offset: 0

N. J. A. Sloane, Jan 23 2020

nonn

Equivalently, this is A334690(n) + 4*n.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 10.

Cf. A255011, A331448.

For the circular analog see A006533, A007678, A007569, A135565.

By Euler's formula, a(n) = A331448(n) - A255011(n) + 1.

a(11)-a(35) from Giovanni Resta, Jan 28 2020

A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, 371100, 466936, 609916, 804504, 1139632, 1288536, 1813288, 2012676, 2536572, 3142008, 3997580, 4230340, 5430444, 6331892, 7360512, 8262568, 10367804

Offset: 0

N. J. A. Sloane, Jan 23 2020, based on a comment from Michael De Vlieger in A255011

nonn

By Euler's formula, a(n) = A331449(n) + A255011(n) - 1.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).

Cf. A255011, A331449.

For the circular analog see A006533, A007678, A007569, A135565.

a(11)-a(29) from Giovanni Resta, Jan 28 2020

A334690 a(n) is the number of points in the interior of the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 29, 245, 801, 2737, 4801, 12265, 19209, 33513, 49537, 87641, 101933, 178413, 220057, 286297, 379033, 551601, 606169, 880217, 951365, 1208965, 1507433, 1947785, 2002573, 2624309, 3055989, 3550317, 3954937, 5068921, 5062033, 6669541, 7081841, 8143061, 9364953, 10296329, 10918073, 13772225, 14842297, 16312317

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 17 2020

nonn

a(n) = A331449(n) - 4*n. Computed from Lars Blomberg's b-file for A331449.

N. J. A. Sloane, Table of n, a(n) for n = 1..52

Cf. A255011, A331448, A331449.

These are the sums of the rows in A334691.

A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

Original entry on oeis.org

1, 20, 8, 1, 204, 32, 8, 0, 1, 616, 152, 20, 8, 4, 0, 1, 2428, 252, 36, 16, 4, 0, 0, 0, 1, 3968, 572, 156, 72, 16, 8, 4, 0, 4, 0, 1, 11164, 900, 120, 52, 16, 8, 4, 0, 0, 0, 0, 0, 1, 16884, 1712, 396, 132, 40, 20, 8, 8, 8, 0, 0, 0, 0, 0, 1, 30116, 2536, 600, 140, 60, 24, 8, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

No formula is known.

			Triangle begins:
1;
20,8,1;
204,32,8,0,1;
616,152,20,8,4,0,1;
2428,252,36,16,4,0,0,0,1;
3968,572,156,72,16,8,4,0,4,0,1;
11164,900,120,52,16,8,4,0,0,0,0,0,1;
16884,1712,396,132,40,20,8,8,8,0,0,0,0,0,1;
30116,2536,600,140,60,24,8,20,8,0,0,0,0,0,0,0,1;
43988,4056,948,312,84,56,52,20,,8,0,0,4,0,0,0,0,0,1;
82016,4660,580,228,48,84,4,4,4,8,4,0,0,0,0,0,0,0,0,0,1;
90088,8504,1840,780,424,128,68,32,32,0,0,8,24,0,0,0,4,0,0,0,0,0,1;
168360,8284,1056,396,128,100,52,12,4,4,4,8,4,0,0,0,0,0,0,0,0,0,0,0,1;
202332,13144,2980,924,256,144,140,60,44,4,0,8,8,8,0,0,4,0,0,0,0,0,0,0,0,0,1;
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.

Cf. A255011, A331449, A334690 (row sums), A334692 (column k=2), A334693 (k=3), A334694-A334699.

A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

Original entry on oeis.org

5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

An equivalent definition: Place n-1 points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of vertices in the resulting planar graph. "In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet. - Scott R. Shannon and N. J. A. Sloane, Nov 05 2023
Equivalently, this is A334697(n) + 4*n.
This is an upper bound on A331449.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Scott R. Shannon, "General position" image for n = 1.
Scott R. Shannon, "General position" image for n = 2.
Scott R. Shannon, "General position" image for n = 3.
Scott R. Shannon, "General position" image for n = 4.
Scott R. Shannon, "General position" image for n = 5.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.

A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* Paolo Xausa, Nov 14 2023 *)

Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.

Edited by N. J. A. Sloane, Nov 13 2023

A334699 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

4, 48, 8, 204, 112, 24, 680, 384, 56, 0, 1660, 1072, 448, 76, 8, 3416, 2392, 400, 56, 0, 0, 6348, 5236, 1840, 504, 40, 0, 0, 11048, 8624, 2800, 408, 24, 0, 0, 0, 17708, 14976, 4960, 1004, 96, 4, 0, 0, 0, 27520, 22200, 7104, 1288, 136, 8, 0, 0, 0, 0, 40388, 35832, 15080, 3808, 488, 60, 0, 0, 0, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

No formula is known.

For further illustrations see A331452.

			Triangle begins:
      4;
     48,     8;
    204,   112,    24;
    680,   384,    56,    0;
   1660,  1072,   448,   76,    8;
   3416,  2392,   400,   56,    0,   0;
   6348,  5236,  1840,  504,   40,   0,  0;
  11048,  8624,  2800,  408,   24,   0,  0, 0;
  17708, 14976,  4960, 1004,   96,   4,  0, 0, 0;
  27520, 22200,  7104, 1288,  136,   8,  0, 0, 0, 0;
  40388, 35832, 15080, 3808,  488,  60,  0, 0, 0, 0, 0;
  57360, 47160, 13856, 2272,  296,  16,  0, 0, 0, 0, 0, 0;
  79108, 73572, 30800, 7788, 1152, 200, 16, 0, 0, 0, 0, 0, 0;
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Image showing the k-sided cells for n = 10.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Cf. A255011 (row sums), A331452, A331449, A334690, A334691, A334695, A334696, A334700.

a(29) and beyond by Scott R. Shannon, Feb 18 2021

A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.

Original entry on oeis.org

1, 50, 363, 1360, 3665, 8106, 15715, 27728, 45585, 70930, 105611, 151680, 211393, 287210, 381795, 498016, 638945, 807858, 1008235, 1243760, 1518321, 1836010, 2201123, 2618160, 3091825, 3627026, 4228875, 4902688, 5653985, 6488490, 7412131, 8431040, 9551553, 10780210, 12123755, 13589136, 15183505, 16914218, 18788835

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

			Scott Shannon's illustration for n=2 shows 29 interior intersection points, of which 20 are simple intersections, 8 are triple intersections, and one (the central point) is a 4-fold intersection. A point where d lines meet is equivalent to C(d,2) simple points. So a(2) = 20*1 + 8*3 + 1*6 = 50.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

Vec(x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5 + O(x^30)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n-4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A334700 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = one-quarter of the number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 12, 2, 51, 28, 6, 170, 96, 14, 0, 415, 268, 112, 19, 2, 854, 598, 100, 14, 0, 0, 1587, 1309, 460, 126, 10, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

This is the triangle in A334699 with all entries divided by 4.
No formula is known.
For illustrations see A331452.

			Triangle begins:
1,
12, 2,
51, 28, 6,
170, 96, 14, 0,
415, 268, 112, 19, 2,
854, 598, 100, 14, 0, 0,
1587, 1309, 460, 126, 10, 0, 0
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2

Cf. A255011, A331452, A331449, A334690, A334691, A334695, A334696, A334699.

A331452 Triangle read by rows: T(n,m) (n >= m >= 1) = number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

4, 16, 56, 46, 142, 340, 104, 296, 608, 1120, 214, 544, 1124, 1916, 3264, 380, 892, 1714, 2820, 4510, 6264, 648, 1436, 2678, 4304, 6888, 9360, 13968, 1028, 2136, 3764, 6024, 9132, 12308, 17758, 22904, 1562, 3066, 5412, 8126, 12396, 16592, 23604, 29374, 38748, 2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line segment. The lines do not extend outside the grid. T(m,n) is the number of regions formed by these lines, and A331453(m,n) and A331454(m,n) give the number of vertices and the number of line segments respectively.

A288187 is a similar sequence, except there every pair of the (m+1)*(n+1) points of the grid (including the interior points) are joined by line segments. The (m,1) (m>=1) and (2,2) entries here and in A288187 are the same, while all other entries are different.

			Triangle begins:
     4;
    16,   56;
    46,  142,  340;
   104,  296,  608,  1120;
   214,  544, 1124,  1916,  3264;
   380,  892, 1714,  2820,  4510,  6264;
   648, 1436, 2678,  4304,  6888,  9360, 13968;
  1028, 2136, 3764,  6024,  9132, 12308, 17758, 22904;
  1562, 3066, 5412,  8126, 12396, 16592, 23604, 29374, 38748;
  2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;
  ...

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Johnny Fonseca, Intersections and Segments, Illustrations for T(n,m) with 2 <= n <= m <= 10, with intersection points shown on the left, and the full structures on the right. Solution to homework problem, Math 640, Rutgers Univ., Feb 11 2020.
Johnny Fonseca, Intersections and Segments, Illustrations for T(n,m) with 2 <= n <= m <= 10, with intersection points shown on the left, and the full structures on the right. Solution to homework problem, Math 640, Rutgers Univ., Feb 11 2020. [Local copy]
Scott R. Shannon, Colored illustration for T(1,1)
Scott R. Shannon, Colored illustration for T(2,1)
Scott R. Shannon, Colored illustration for T(3,1)
Scott R. Shannon, Colored illustration for T(4,1)
Scott R. Shannon, Colored illustration for T(5,1)
Scott R. Shannon, Colored illustration for T(6,1)
Scott R. Shannon, Colored illustration for T(7,1)
Scott R. Shannon, Colored illustration for T(8,1)
Scott R. Shannon, Colored illustration for T(9,1)
Scott R. Shannon, Colored illustration for T(10,1)
Scott R. Shannon, Colored illustration for T(11,1)
Scott R. Shannon, Colored illustration for T(12,1)
Scott R. Shannon, Colored illustration for T(13,1)
Scott R. Shannon, Colored illustration for T(14,1)
Scott R. Shannon, Colored illustration for T(15,1)
Scott R. Shannon, Colored illustration for T(2,2)
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(4,2)
Scott R. Shannon, Colored illustration for T(5,2)
Scott R. Shannon, Colored illustration for T(6,2)
Scott R. Shannon, Colored illustration for T(9,2)
Scott R. Shannon, Colored illustration for T(9,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(10,2)
Scott R. Shannon, Colored illustration for T(10,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(4,3)
Scott R. Shannon, Colored illustration for T(5,3)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(9,3)
Scott R. Shannon, Colored illustration for T(11,3) [The top of the figure has been modified]
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(5,4)
Scott R. Shannon, Colored illustration for T(6,4)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(6,5)
Scott R. Shannon, Colored illustration for T(6,6)
Scott R. Shannon, Colored illustration for T(6,6) (another version)
Scott R. Shannon, Colored illustration for T(7,7)
Scott R. Shannon, Colored illustration for T(10,7)
Scott R. Shannon, Data underlying this triangle and A331453, A331454 [Includes numbers of polygonal regions with each number of edges.]
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles
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, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)

The first column is A306302, the main diagonal is A255011.
The second column is A331766.
See A333274 for the classification of vertices by valency.
Cf. A288187, A331453, A331454, A333286, A333287, A333288.

A355798 Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

1, 4, 24, 104, 316, 712, 1588, 2816, 4940, 7672, 12444, 16840, 25968, 34088, 46260, 61048, 82792, 98984, 133032, 156072, 196236, 239048, 298292, 334032, 417072, 483856, 570200, 649816, 786412, 850000, 1037628, 1145424, 1311536, 1485880, 1677660, 1828360, 2158192, 2357376, 2623604, 2852688

Offset: 1

Scott R. Shannon, Jul 17 2022

nonn

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 16.
Scott R. Shannon, Alternative version for n = 15. The lines joining the vertices that are exactly opposite are omitted, forming a more carpet-like image. The number of regions formed, 42224, is less than the sequence version, although more 5,6,7 and 8-gon regions are present.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 20-21.

Cf. A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

a(n) = A355800(n) - A355799(n) + 1 by Euler's formula.

cp's OEIS Frontend

A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A334690 a(n) is the number of points in the interior of the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

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A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

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A334699 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.

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A334700 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = one-quarter of the number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A331452 Triangle read by rows: T(n,m) (n >= m >= 1) = number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

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