cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A331758 A331757(n)/2.

Original entry on oeis.org

4, 14, 40, 89, 186, 327, 562, 891, 1362, 1957, 2790, 3813, 5176, 6795, 8770, 11105, 14020, 17335, 21380, 25981, 31306, 37247, 44254, 52021, 60956, 70767, 81832, 93971, 107818, 122745, 139630, 158011, 178228, 199949, 223806, 249349, 277676, 307887
Offset: 1

Views

Author

N. J. A. Sloane, Feb 04 2020

Keywords

Crossrefs

Cf. A331757.

Programs

  • Mathematica
    Table[(n^2 + 4n + 1 + Sum[Sum[(2 * Boole[GCD[i, j] == 1] - Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}]) / 2, {n, 1, 38}] (* Joshua Oliver, Feb 05 2020 *)

A306302 Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).

Original entry on oeis.org

0, 4, 16, 46, 104, 214, 380, 648, 1028, 1562, 2256, 3208, 4384, 5924, 7792, 10052, 12744, 16060, 19880, 24486, 29748, 35798, 42648, 50648, 59544, 69700, 80992, 93654, 107596, 123374, 140488, 159704, 180696, 203684, 228624, 255892, 285152, 317400, 352096, 389576
Offset: 0

Views

Author

Paarth Jain, Feb 05 2019

Keywords

Comments

Assuming that the rectangles have vertices at (k,0) and (k,1), k=0..n, the projective map (x,y) -> ((1-y)/(x+1),y/(x+1)) maps their partition to the partition of the right isosceles triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. - Max Alekseyev, Apr 10 2019
The figure is made up of A324042 triangles and A324043 quadrilaterals. - N. J. A. Sloane, Mar 03 2020

Links

Crossrefs

See A331755 for the number of vertices, A331757 for the number of edges.
Cf. A007678, A108914, A114043, A324042, A324043, A333286, A333287, A333288, A334694.
A column of A288187. See A288177 for additional references.
Also a column of A331452 and A356790.
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. - N. J. A. Sloane, Feb 04 2020

Programs

  • Maple
    # Maple from N. J. A. Sloane, Mar 04 2020, starting at n=1:  First define z(n) = A115004
z := proc(n)
    local a, b, r ;
    r := 0 ;
    for a from 1 to n do
    for b from 1 to n do
        if igcd(a, b) = 1 then
            r := r+(n+1-a)*(n+1-b);
        end if;
    end do:
    end do:
    r ;
end proc:
a := n-> z(n)+n^2+2*n;
[seq(a(n), n=1..50)];
  • Mathematica
    z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[0] = 0;
a[n_] := z[n] + n^2 + 2n;
a /@ Range[0, 40] (* Jean-François Alcover, Mar 24 2020 *)
  • Python
    from sympy import totient
def A306302(n): return 2*n*(n+1) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 16 2021

Formula

a(n) = n + (A114043(n+1) - 1)/2, conjectured by N. J. A. Sloane, Feb 07 2019; proved by Max Alekseyev, Apr 10 2019
a(n) = n + A115005(n+1) = n + A141255(n+1)/2. - Max Alekseyev, Apr 10 2019
a(n) = A324042(n) + A324043(n). - Jinyuan Wang, Mar 19 2020
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n. - N. J. A. Sloane, Apr 11 2020
a(n) = 2n(n+1) + Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

Extensions

a(6)-a(20) from Robert Israel, Feb 07 2019
Edited and more terms added by Max Alekseyev, Apr 10 2019
a(0) added by N. J. A. Sloane, Feb 04 2020

A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

Original entry on oeis.org

8, 28, 92, 80, 240, 360, 178, 508, 604, 860, 372, 944, 1040, 1320, 1792, 654, 1548, 1652, 1956, 2452, 3124, 1124, 2520, 2640, 2968, 3488, 4184, 5256, 1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188, 2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304, 3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 03 2020

Keywords

Comments

See A331457 and A331776 for further illustrations.
There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

Examples

			Triangle begins:
[8],
[28, 92],
[80, 240, 360],
[178, 508, 604, 860],
[372, 944, 1040, 1320, 1792],
[654, 1548, 1652, 1956, 2452, 3124],
[1124, 2520, 2640, 2968, 3488, 4184, 5256],
[1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188],
[2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304],
[3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460],
...

Links

Crossrefs

Cf. A331457, A331757, A331765, A331776, A332599, A332610, A332611.
The main diagonal is A332597.

Formula

Column 1 is A331757, for which there is an explicit formula.
Column 2 is A331765, for which no formula is known.
For m >= n >= 3, T(m,n) = (3*A332610(m,n)+4*A332611(m,n)+4*m+4*n-8)/2, and both A332610 and A332611 have explicit formulas.

Extensions

More terms from N. J. A. Sloane, Mar 13 2020

A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 240, 596, 178, 508, 1028, 1936, 372, 944, 2004, 3404, 6020, 654, 1548, 3018, 4962, 8064, 11088, 1124, 2520, 4808, 7734, 12708, 17022, 26260, 1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144, 2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296, 3914, 7528, 12720, 19428, 29914, 37848, 54612, 67590, 86906, 107832
Offset: 1

Views

Author

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

Keywords

Comments

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 (of finite length). The lines do not extend outside the grid. T(m,n) is the number of line segments formed when these lines intersect each other, and A331452(m,n) and A331453(m,n) give the number of regions and the number of vertices respectively.
For illustrations see the links in A331452.

Examples

			Triangle begins:
8,
28, 92,
80, 240, 596,
178, 508, 1028, 1936,
372, 944, 2004, 3404, 6020,
654, 1548, 3018, 4962, 8064, 11088,
1124, 2520, 4808, 7734, 12708, 17022, 26260,
1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144,
2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296,
...

Links

Crossrefs

The main diagonal is A331448.
The first two columns are A331757 and A331765.
Cf. A288187, A331452, A331453.

A333274 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of vertices in the graph at which k polygons meet.

Original entry on oeis.org

4, 0, 1, 0, 4, 8, 0, 1, 0, 0, 28, 4, 2, 0, 1, 0, 0, 54, 4, 14, 0, 2, 0, 1, 0, 0, 124, 0, 22, 8, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 4, 20, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 26, 12, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 4, 26, 0, 2, 0, 2, 0, 2, 0, 1
Offset: 1

Views

Author

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

Keywords

Comments

For vertices not on the boundary, the number of polygons meeting at a vertex is simply the degree (or valency) of that vertex.
Row sums are A331755.
Sum_k k*T(n,k) gives A333276.
See A333275 for the degrees of the non-boundary vertices.
Row n is the sum of [0, 0, ..., 0 (n-1 0's), 4, 2*n-2, 0, 0, ..., 0 (n 0's)] and row n of A333275.

Examples

			Led d denote the number of polygons meeting at a vertex (except for boundary points, d is the degree of the vertex).
For n=2, the 4 corners have d=3, and on the center line there are 2 vertices with d=4 and 1 with d=6. In the interiors of each of the two squares there are 3 points with d=4.
So in total there are 4 points with d=3, 8 with d=4, and 1 with d=6. So row 2 of the triangle is [0, 4, 8, 0, 1].
The triangle begins:
4,0,1,
0,4,8,0,1,
0,0,28,4,2,0,1,
0,0,54,4,14,0,2,0,1,
0,0,124,0,22,8,2,0,2,0,1,
0,0,214,0,32,4,20,0,2,0,2,0,1;
0,0,382,0,50,0,26,12,2,0,2,0,2,0,1;
0,0,598,0,102,0,18,4,26,0,2,0,2,0,2,0,1;
0,0,950,0,126,0,32,0,30,16,2,0,2,0,2,0,2,0,1;
0,0,1334,0,198,0,62,0,20,4,32,0,2,0,2,0,2,0,2,0,1;
0,0,1912,0,286,0,100,0,10,0,34,20,2,0,2,0,2,0,2,0,2,0,1;
0,0,2622,0,390,0,118,0,38,0,22,4,38,0,2,0,2,0,2,0,2,0,2,0,1;
0,0,3624,0,510,0,136,0,74,0,10,0,38,24,2,0,2,0,2,0,2,0,2,0,2,0,1;
0,0,4690,0,742,0,154,0,118,0,10,0,24,4,44,0,2,0,2,0,2,0,2,0,2,0,2,0,1;

Links

Crossrefs

Cf. A306302, A331755, A331757, A331452, A333275, A333276, A333277.

A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

Original entry on oeis.org

0, 0, 1, 0, 0, 6, 0, 1, 0, 0, 24, 0, 2, 0, 1, 0, 0, 54, 0, 8, 0, 2, 0, 1, 0, 0, 124, 0, 18, 0, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 0, 10, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 22, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 0, 12, 0, 2, 0, 2, 0, 2, 0, 1
Offset: 1

Views

Author

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

Keywords

Comments

The number of polygons meeting at a non-boundary vertex is simply the degree (or valency) of that vertex.
Row sums are A159065.
Sum_k k*T(n,k) gives A333277.
See A333274 for the degrees if the boundary vertices are included.
T(n,k) = 0 if k is odd. But the triangle includes those zero entries because this is used to construct A333274.

Examples

			Led d denote the number of polygons meeting at a vertex.
For n=2, in the interiors of each of the two squares there are 3 points with d=4, and the center point has d=6.
So in total there are 6 points with d=4 and 1 with d=6. So row 2 of the triangle is [0, 0, 6, 0, 1].
The triangle begins:
0,0,1,
0,0,6,0,1,
0,0,24,0,2,0,1,
0,0,54,0,8,0,2,0,1,
0,0,124,0,18,0,2,0,2,0,1,
0,0,214,0,32,0,10,0,2,0,2,0,1,
0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,
0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
...
If we leave out the uninteresting zeros, the triangle begins:
[1]
[6, 1]
[24, 2, 1]
[54, 8, 2, 1]
[124, 18, 2, 2, 1]
[214, 32, 10, 2, 2, 1]
[382, 50, 22, 2, 2, 2, 1]
[598, 102, 18, 12, 2, 2, 2, 1]
[950, 126, 32, 26, 2, 2, 2, 2, 1]
[1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]
[1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]
[2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]
... - _N. J. A. Sloane_, Jul 27 2020

Links

Crossrefs

Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694.

Extensions

a(36) and beyond from Lars Blomberg, Jun 17 2020

A333279 Column 2 of triangle in A288187.

Original entry on oeis.org

16, 56, 176, 388, 822, 1452, 2516, 3952, 6060, 8736, 12492, 17040, 23102, 30280, 39234, 49688, 62730, 77556, 95642, 115992, 139874, 166560, 197992, 232600, 272574, 316460, 366390, 420792, 482748, 549516, 624962, 706436, 796766, 893844, 1001074, 1115428
Offset: 1

Views

Author

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

Keywords

Comments

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.
The maximum number of edges over all chambers is 4 for 1 <= n <= 4 and 5 for 5 <= n <= 160. - Lars Blomberg, May 23 2021

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A333280 Column 2 of triangle in A333278.

Original entry on oeis.org

28, 92, 296, 652, 1408, 2470, 4312, 6774, 10428, 14992, 21492, 29328, 39876, 52184, 67616, 85588, 108192, 133674, 164992, 200158, 241560, 287428, 341768, 401472, 470764, 546230, 632404, 726170, 833420, 948550, 1079204, 1220054, 1376552, 1543742, 1729000
Offset: 1

Views

Author

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

Keywords

Comments

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.
See A333279 for illustrations.

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A333281 Column 2 of triangle in A288180.

Original entry on oeis.org

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581
Offset: 1

Views

Author

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

Keywords

Comments

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.
See A333279 for illustrations.

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

1, 3, 3, 5, 8, 5, 7, 15, 15, 7, 9, 24, 28, 24, 9, 11, 35, 47, 47, 35, 11, 13, 48, 69, 80, 69, 48, 13, 15, 63, 97, 119, 119, 97, 63, 15, 17, 80, 128, 170, 178, 170, 128, 80, 17, 19, 99, 165, 225, 257, 257, 225, 165, 99, 19, 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
Offset: 1

Views

Author

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

Keywords

Comments

The case m=n (the main diagonal) is dealt with in A331757. A306302 has illustrations for the diagonal case for m = 1 to 15.
Also A335678 has colored illustrations for many values of m and n.

Examples

			The initial rows of the array are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ...
3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
5, 15, 28, 47, 69, 97, 128, 165, 205, 251, 300, 355, ...
7, 24, 47, 80, 119, 170, 225, 292, 365, 448, 537, 638, ...
9, 35, 69, 119, 178, 257, 340, 443, 555, 683, 819, 975, ...
11, 48, 97, 170, 257, 372, 493, 644, 809, 998, 1197, 1426, ...
13, 63, 128, 225, 340, 493, 654, 857, 1078, 1331, 1595, 1901, ...
15, 80, 165, 292, 443, 644, 857, 1124, 1415, 1748, 2095, 2498, ...
17, 99, 205, 365, 555, 809, 1078, 1415, 1782, 2203, 2640, 3149, ...
19, 120, 251, 448, 683, 998, 1331, 1748, 2203, 2724, 3265, 3896, ...
21, 143, 300, 537, 819, 1197, 1595, 2095, 2640, 3265, 3914, 4673, ...
...
The initial antidiagonals are:
1
3, 3
5, 8, 5
7, 15, 15, 7
9, 24, 28, 24, 9
11, 35, 47, 47, 35, 11
13, 48, 69, 80, 69, 48, 13
15, 63, 97, 119, 119, 97, 63, 15
17, 80, 128, 170, 178, 170, 128, 80, 17
19, 99, 165, 225, 257, 257, 225, 165, 99, 19
21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
23, 143, 251, 365, 443, 493, 493, 443, 365, 251, 143, 23
...

Links

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331757.
Cf. A114999, A331762.

Formula

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = 2*A114999(m-1,n-1) - A331762(m-1,n-1) + m*n + m + n - 2 for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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