A306302 -id:A306302 - cp's OEIS frontend

A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).

0, 0, 2, 11, 32, 77, 148, 268, 442, 691, 1018, 1472, 2036, 2780, 3686, 4786, 6100, 7724, 9598, 11863, 14454, 17437, 20818, 24772, 29172, 34200, 39794, 46071, 52986, 60817, 69314, 78860, 89292, 100720, 113122, 126686, 141244, 157294, 174566, 193228, 213172, 234954, 258058, 283189, 309946, 338473, 368782, 401516, 436040

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

nonn

Related to the number of linear dichotomies on a square grid.

A331771(n) = 8*a(n) + 4*n*(n-1) + 4*(n-1)^2.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Jack Koplowitz, Michael Lindenbaum, and A. Bruckstein, The number of digital straight lines on an NxN grid, IEEE Transactions on Information Theory 36.1 (1990): 192-197. (See I_1.)

The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. The present sequence and A331771 could be added to this list.

I1 := proc(n) local a, i, j; a:=0;
for i from 2 to n-1 do for j from 1 to i-1 do
if igcd(i,j)=1 then a := a+(n-i)*(n-j); fi; od; od; a; end;
[seq(I1(n),n=1..40)];

a(n) = sum(i=2, n-1, sum(j=1, i-1, if (gcd(i,j)==1, (n-i)*(n-j)))); \\ Michel Marcus, Mar 14 2020

from sympy import totient
def A332612(n): return sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))//2 # Chai Wah Wu, Aug 17 2021

a(n) = (Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i))/2. - Chai Wah Wu, Aug 17 2021

A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 296, 872, 178, 652, 1922, 4344, 372, 1408, 4256, 9738, 21284, 654, 2470, 7466, 16978, 36922, 64172, 1124, 4312, 13112, 29874, 64800, 113494, 200028, 1782, 6774, 20812, 47402, 103116, 181484, 319516, 509584, 2724, 10428, 31776, 72398, 158352, 279070, 490396, 782096, 1199428

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

T(n,m) = A288180(n,m)+A288187(n,m)-1 (Euler).

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

			Triangle begins:
8,
28, 92,
80, 296, 872,
178, 652, 1922, 4344,
372, 1408, 4256, 9738, 21284,
654, 2470, 7466, 16978, 36922, 64172,
...

Hugo Pfoertner, Illustration of intersection points up to 6 X 6.

Cf. A288180.
For column 1 see A331757. For column 2 see A333279, A333280, A333281.
Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

A335056 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n) is the number of vertices inside the k-th rectangle.

Original entry on oeis.org

1, 3, 3, 5, 11, 5, 7, 19, 19, 7, 9, 29, 43, 29, 9, 11, 37, 61, 61, 37, 11, 13, 47, 83, 105, 83, 47, 13, 15, 57, 103, 143, 143, 103, 57, 15, 17, 69, 125, 183, 211, 183, 125, 69, 17, 19, 81, 143, 215, 267, 267, 215, 143, 81, 19, 21, 95, 167, 253, 329, 369, 329, 253, 167, 95, 21, 23, 109, 189, 289, 385, 455, 455, 385, 289, 189, 109, 23

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 21 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			Triangle begins:
1;
3, 3;
5, 11, 5;
7, 19, 19, 7;
9, 29, 43, 29, 9;
11, 37, 61, 61, 37, 11;
13, 47, 83, 105, 83, 47, 13;
15, 57, 103, 143, 143, 103, 57, 15;
17, 69, 125, 183, 211, 183, 125, 69, 17;
19, 81, 143, 215, 267, 267, 215, 143, 81, 19;
21, 95, 167, 253, 329, 369, 329, 253, 167, 95, 21;
23, 109, 189, 289, 385, 455, 455, 385, 289, 189, 109, 23;
25, 125, 215, 331, 451, 551, 597, 551, 451, 331, 215, 125, 25;

Scott R. Shannon, Image for n = 2 showing the count of the vertices.
Scott R. Shannon, Image for n = 3 showing the count of the vertices.
Scott R. Shannon, Image for n = 5 showing the count of the vertices.
Scott R. Shannon, Image for n = 9 showing the count of the vertices.
Scott R. Shannon, Image for n = 12 showing the count of the vertices.

Cf. A335074, A159065, A331755, A333288, A306302.

Row sum n + Row sum A335074(n) = A159065(n).

A335074 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n-1) is the number of vertices on the edge separating rectangles k and k+1.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 7, 7, 7, 9, 9, 7, 9, 9, 11, 13, 11, 11, 13, 11, 13, 15, 17, 11, 17, 15, 13, 15, 19, 19, 19, 19, 19, 19, 15, 17, 21, 25, 21, 19, 21, 25, 21, 17, 19, 25, 29, 29, 23, 23, 29, 29, 25, 19, 21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21, 23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23

Offset: 2

Scott R. Shannon and N. J. A. Sloane, May 22 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			Triangle begins:
1;
3, 3;
5, 3, 5;
7, 7, 7, 7;
9, 9, 7, 9, 9;
11, 13, 11, 11, 13, 11;
13, 15, 17, 11, 17, 15, 13;
15, 19, 19, 19, 19, 19, 19, 15;
17, 21, 25, 21, 19, 21, 25, 21, 17;
19, 25, 29, 29, 23, 23, 29, 29, 25, 19;
21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21;
23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23;

Scott R. Shannon, Image for n = 3 showing the count of the vertices.
Scott R. Shannon, Image for n = 4 showing the count of the vertices.
Scott R. Shannon, Image for n = 7 showing the count of the vertices.
Scott R. Shannon, Image for n = 10 showing the count of the vertices.
Scott R. Shannon, Image for n = 12 showing the count of the vertices.

Cf. A335056, A159065, A331755, A333288, A306302.

Row sum n + Row sum A335056(n) = A159065(n).

A356790 Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of regions formed by straight line segments when connecting the k-1 points along the top side of a rectangle to each of the k-1 points along the bottom side that divide these sides into k equal parts, along with straight lines that directly connect the n-1 points along the left side to the diametrically opposite point on the right side that divide these sides into n equal parts.

Original entry on oeis.org

1, 2, 2, 6, 4, 3, 18, 10, 6, 4, 48, 24, 16, 8, 5, 106, 56, 34, 20, 10, 6, 216, 116, 70, 44, 26, 12, 7, 382, 228, 134, 84, 58, 30, 14, 8, 650, 396, 250, 152, 112, 60, 36, 16, 9, 1030, 666, 422, 272, 190, 112, 78, 40, 18, 10, 1564, 1048, 696, 448, 320, 196, 150, 84, 46, 20, 11

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 04 2022

			The table begins:
1,  2,  6,  18,  48,  106, 216, 382, 650,  1030, 1564, 2258, 3210, 4386, 5926, ...
2,  4,  10, 24,  56,  116, 228, 396, 666,  1048, 1584, 2280, 3234, 4412, 5954, ...
3,  6,  16, 34,  70,  134, 250, 422, 696,  1082, 1622, 2322, 3280, 4462, 6008, ...
4,  8,  20, 44,  84,  152, 272, 448, 726,  1116, 1660, 2364, 3326, 4512, 6062, ...
5,  10, 26, 58,  112, 190, 320, 506, 794,  1194, 1748, 2462, 3434, 4630, 6190, ...
6,  12, 30, 60,  112, 196, 326, 512, 800,  1200, 1754, 2468, 3440, 4636, 6196, ...
7,  14, 36, 78,  150, 258, 418, 626, 936,  1358, 1934, 2670, 3664, 4882, 6464, ...
8,  16, 40, 84,  152, 256, 414, 632, 942,  1364, 1940, 2676, 3670, 4888, 6470, ...
9,  18, 46, 94,  172, 290, 468, 710, 1050, 1490, 2084, 2838, 3850, 5086, 6686, ...
10, 20, 50, 104, 188, 304, 480, 720, 1060, 1516, 2112, 2868, 3882, 5120, 6722, ...
11, 22, 56, 118, 218, 366, 586, 878, 1280, 1794, 2454, 3258, 4320, 5606, 7256, ...
12, 24, 60, 120, 208, 336, 518, 764, 1114, 1580, 2204, 2992, 4020, 5272, 6888, ...
.
.
See the attached table for further terms.

Scott R. Shannon, Table for n=1..40, k=1..40.
Scott R. Shannon, Image of T(3,4) = 34.
Scott R. Shannon, Image of T(5,7) = 320.
Scott R. Shannon, Image of T(8,6) = 256.
Scott R. Shannon, Image of T(11,14) = 5606.
Scott R. Shannon, Image of T(16,16) = 9964.

Cf. A146951, A355798, A306302, A290131, A331452, A355902.

T(1,k) = A306302(k-2) + 2, k >= 2.
T(2,k) = 2*A355902(k-2) + 4 = A306302(k-2) + 2*k, k >= 2.
T(n,1) = n.
T(n,2) = 2n.
T(n,3) = A146951(n).

A335861 Number of regions in a Y-shaped polygon with equal arms of length n (see the Comments for definition).

Original entry on oeis.org

1, 70, 349, 916, 1474, 2296, 3412, 4978, 7042, 9748, 13132, 17506, 22786, 29410, 37288, 46630, 57574

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 22 2020

This polygon consists of a central equilateral triangle with a line of n adjacent squares connected to each of its three edges. This gives the polygon a total of one triangle, 3n squares, and 6n+3 vertices. Join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the polygon. The sequence gives the number of regions in the resulting figure.

			a(0) = 1. There is one region in an equilateral triangle with no other polygons.
a(1) = 70. With one square adjacent to each of the triangles sides the resulting line segments form 48 triangles, twelves 4-gons, nine 5-gons, and one 6-gon. This gives a total of 70 regions. See the first linked image.

Scott R. Shannon, Image for the figure with edge-count coloring for n=1.
Scott R. Shannon, Image for the figure with edge-count coloring for n=2.
Scott R. Shannon, Image for the figure with edge-count coloring for n=3.
Scott R. Shannon, Image for the figure with edge-count coloring for n=4.
Scott R. Shannon, Image for the figure with edge-count coloring for n=5.
Scott R. Shannon, Image for the figure with edge-count coloring for n=6.

Cf. A337790 (number of vertices), A331456, A331452, A306302, A092867, A007678.

A336731 Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).

Original entry on oeis.org

4, 0, 0, 14, 8, 0, 20, 48, 4, 60, 80, 28, 68, 224, 68, 148, 368, 124, 224, 616, 268, 336, 1008, 420, 384, 1672, 648, 712, 2208, 972, 972, 3120, 1464, 1300, 4304, 1996, 1496, 6040, 2788, 2044, 7936, 3580, 2612, 10224, 4672, 3540, 12656, 5980, 4224, 16104, 7676, 5484, 19648, 9500

Offset: 1

Scott R. Shannon, Aug 02 2020

For a row of n adjacent rectangles the only polygons formed when dividing all possible rectangles along their diagonals are 3-gons (triangles) and 4-gons (quadrilaterals). Hence the only possible edge-sharing contacts are 3-gons with 3-gons, 3-gons with 4-gons, and 4-gons with 4-gons. This sequence lists the number of these three possible combinations for a row of n adjacent rectangles. Note that the edges along the outside of the n adjacent rectangles are not counted as they are only in one n-gon.

These are graphs T(1,n) described in A331452. - N. J. A. Sloane, Aug 03 2020

			a(1) = 4, a(2) = 0, a(3) = 0. A single rectangle divided along its diagonals consists of four 3-gons, four edges, and no 4-gons. Therefore there are only four 3-gon-to-3-gon contacts. See the link image for n = 1.
a(4) = 14, a(5) = 8, a(6) = 0. Two adjacent rectangles divided along all diagonals consists of fourteen 3-gons and two 4-gons. The two 4-gons are separated and thus share all their edges, eight in total, with 3-gons. There are fourteen pairs of 3-gon-to-3-gon contacts. See the link image for n = 2.
a(7) = 20, a(8) = 48, a(9) = 4. Three adjacent rectangles divided along all diagonals consists of thirty-two 3-gons and fourteen 4-gons. There are two groups of three adjacent 4-gons, so there are four 4-gons-to-4-gon contacts. These, along with the other 4-gons, share 48 edges with 3-gons. There are also twenty 3-gon-to-3-gon contacts. See the link image for n = 3.
.
The table begins:
4,0,0;
14,8,0;
20,48,4;
60,80,28;
68,224,68;
148,368,124;
224,616,268;
336,1008,420;
384,1672,648;
712,2208,972;
972,3120,1464;
1300,4304,1996;
1496,6040,2788;
2044,7936,3580;
2612,10224,4672;
3540,12656,5980;
4224,16104,7676;
5484,19648,9500;
6568,24216,11936;
7836,29616,14468;
See A306302 for a count of the regions and images for other values of n.

Scott R. Shannon, Image of the rectangles for n = 1.
Scott R. Shannon, Image of the rectangles for n = 2.
Scott R. Shannon, Image of the rectangles for n = 3.
Scott R. Shannon, Image of the rectangles for n = 4.

Cf. A306302, A331452, A331755, A331757, A333288.

Sum of row t = A331757(t) - 2(t + 1).

A337790 Number of vertices in a Y-shaped polygon with equal arms of length n (see the Comments in A335861 for definition).

Original entry on oeis.org

3, 57, 306, 837, 1335, 2073, 3033, 4395, 6147, 8469, 11253, 14907, 19263, 24819, 31197, 38823, 47619

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 22 2020

See A335861 for a definition of the polygon and images for the number of regions.

			a(0) = 3. A single triangle with no other polygons has three vertices.
a(1) = 57. With one square adjacent to each of the triangles sides the resulting line segments form 51 vertices shared by four polygons, 3 vertices shared by six polygons, and 3 vertices shared by seven polygons. This gives a total of 57 vertices. See the first linked image.

Scott R. Shannon, Image for the figure showing the vertex counts for n=1.
Scott R. Shannon, Image for the figure showing the vertex counts for n=2.
Scott R. Shannon, Image for the figure showing the vertex counts for n=3.
Scott R. Shannon, Image for the figure showing the vertex counts for n=4.
Scott R. Shannon, Image for the figure showing the vertex counts for n=5.
Scott R. Shannon, Image for the figure showing the vertex counts for n=6.

Cf. A335861 (number of regions), A331456, A331452, A306302, A092867, A007678.

A355902 Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.

Original entry on oeis.org

0, 3, 10, 26, 56, 112, 196, 331, 522, 790, 1138, 1615, 2204, 2975, 3910, 5041, 6388, 8047, 9958, 12262, 14894, 17920, 21346, 25347, 29796, 34875, 40522, 46854, 53826, 61716, 70274, 79883, 90380, 101875, 114346, 127981, 142612, 158737, 176086, 194827, 214852, 236717, 259906, 285124, 311970, 340588, 370990, 403819, 438440, 475556

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 05 2022

nonn

Note that this figure can be obtained by drawing an "equatorial" line through the middle of the strip of n adjacent rectangles in A306302. This cuts each of the 2n "equatorial" cells in A306302 in two. It follows that 2*a(n) = A306302(n) + 2*n, i.e. that a(n) = A306302(n)/2 + n. Note that there is an explicit formula for A306302(n) in terms of n. - Scott R. Shannon, Sep 06 2022.

This means the present sequence is one more member of the large class of sequences which are essentially the same as A115004 (see Cross-References). - N. J. A. Sloane, Sep 06 2022

Scott R. Shannon, Image for a(4) = 56. Note in this and other images the entire 2xn array is shown so the number of cells is twice a(n).
Scott R. Shannon, Image for a(6) = 196.
Scott R. Shannon, Image for a(10) = 1138.
Scott R. Shannon, Image for a(15) = 5041.
N. J. A. Sloane, Illustration for a(2) = 10.
N. J. A. Sloane, Illustration for a(3) = 26.

Cf. A356790, A306302, A355798, A290131, A331452.

The following nine sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n; A355902(n) = n + A306302(n)/2. - N. J. A. Sloane, Sep 06 2022

a(n) = A356790(2,n+2)/2 - 2.

A337640 a(n) = one-half of the number of cells in the central rectangle of the graph described in row 2n+1 of A333288.

Original entry on oeis.org

2, 11, 35, 80, 155, 266, 422, 626, 890, 1223, 1625, 2108, 2678, 3341, 4109, 4988, 5990, 7106, 8348, 9734, 11264, 12953, 14801, 16820, 19019, 21389, 23957, 26717, 29663, 32834, 36230, 39860, 43712, 47795, 52139, 56726, 61598, 66746, 72152, 77837

Offset: 0

N. J. A. Sloane, Sep 17 2020

nonn

This is based on Lars Blomberg's data in A333288.
A333288 is a triangular array read by rows. a(n) is the central term in row 2n+1 of that triangle, divided by 2.
See A331452 for further illustrations.
It would be nice to have a formula for this sequence. It is possible that focusing on the points (n, a(n)) where 2n+1 is a prime might lead to a simpler formula.

Scott R. Shannon, Colored illustration for a(0): there are 4 regions, so a(0) = 2.
Scott R. Shannon, Colored illustration for a(1): the central square has 22 regions, so a(1) = 11.
Scott R. Shannon, Colored illustration for a(2): the central square has 70 regions, so a(2) = 35.
Scott R. Shannon, Colored illustration for a(3): the central square has 160 regions, so a(3) = 80.
Scott R. Shannon, Colored illustration for a(4): the central square has 310 regions, so a(4) = 155.

Cf. A306302, A331452, A333288.

cp's OEIS Frontend

A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).

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A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

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A335056 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n) is the number of vertices inside the k-th rectangle.

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A335074 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n-1) is the number of vertices on the edge separating rectangles k and k+1.

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A335861 Number of regions in a Y-shaped polygon with equal arms of length n (see the Comments for definition).

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A336731 Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).

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A337790 Number of vertices in a Y-shaped polygon with equal arms of length n (see the Comments in A335861 for definition).

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A355902 Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.

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A337640 a(n) = one-half of the number of cells in the central rectangle of the graph described in row 2n+1 of A333288.