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

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

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

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 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.

Examples

			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;
  ...

References

  • 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.

Links

Crossrefs

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.

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

A018808 Number of lines through at least 2 points of an n X n grid of points.

Original entry on oeis.org

0, 0, 6, 20, 62, 140, 306, 536, 938, 1492, 2306, 3296, 4722, 6460, 8830, 11568, 14946, 18900, 23926, 29544, 36510, 44388, 53586, 63648, 75674, 88948, 104374, 121032, 139966, 160636, 184466, 209944, 239050, 270588, 305478, 342480, 383370, 427020
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A222267 (lines defined by n X n X n grid of points).
A288187 is the main entry for these graphs.
Cf. A331780.

Programs

  • Mathematica
    L[0]=0; L1[1]=0; R1[1]=0;
L[n_]:=L[n]=2*L1[n]-L[n-1]+R1[n]
L1[n_]:=L1[n]=2*L[n-1]-L1[n-1]+R2[n]
R1[n_]:=R1[n]=R1[n-1]+4*(EulerPhi[n-1]-e[n])
e[n_]:=If[Mod[n,2]==0,0,EulerPhi[(n-1)/2]]
R2[n_]:= If[Mod[n,2]==0,(n-1)*EulerPhi[n-1], If[Mod[n,4]==1,(n-1)*EulerPhi[n-1]/2,0]]
Table[L[n],{n,0,37}] (* Seppo Mustonen, Apr 25 2009 *)

Formula

(1/2) * (f(n, 1) - f(n, 2)) where f(n, k) = Sum ((n - |x|)(n - |y|)); -n < x < n, -n < y < n, (x, y)=k.
(1/2) * (f(n, 1) - f(n, 2)) where f(n, k) = Sum ((n - |kx|)(n - |ky|)); -n < kx < n, -n < ky < n, (x, y)=1. - Seppo Mustonen, Apr 18 2009
a(0) = L(0,1) = R1(0) = 0, a(n) = L(n,n) = 2L(n-1,n) - L(n-1,n-1) + R1(n), L(n-1,n) = 2L(n-1,n-1) - L(n-2,n-1) + R2(n), R1(n) = R1(n-1) + 4(phi(n-1) - e(n)), e(n)=0, n even, e(n) = phi((n-1)/2), n odd, R2(n) = (n-1)phi(n-1), n even, R2(n)=(n-1)phi(n-1)/2, n=1 mod 4, R2(n)=0, n=3 mod 4. - Seppo Mustonen, Apr 25 2009
a(n) = 2 * A331780(n). - Alois P. Heinz, Jun 05 2023

A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices 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

5, 13, 37, 35, 99, 257, 75, 213, 421, 817, 159, 401, 881, 1489, 2757, 275, 657, 1305, 2143, 3555, 4825, 477, 1085, 2131, 3431, 5821, 7663, 12293, 755, 1619, 2941, 4817, 7477, 9913, 15037, 19241, 1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549, 1659, 3257, 5603, 8637, 13689, 16953, 25125, 30779, 39857, 49577
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 vertices in the resulting diagram, and A331452(m,n) and A331454(m,n) give the number of regions and the number of line segments respectively.
For illustrations see the links in A331452.

Examples

			Triangle begins:
5,
13, 37,
35, 99, 257,
75, 213, 421, 817,
159, 401, 881, 1489, 2757,
275, 657, 1305, 2143, 3555, 4825,
477, 1085, 2131, 3431, 5821, 7663, 12293,
755, 1619, 2941, 4817, 7477, 9913, 15037, 19241,
1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549,
...

Links

Crossrefs

The main diagonal is A331449.
The first two columns are A331755 and A331763.
Cf. A288187, A331452, A331454.

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.

A288177 Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.

Original entry on oeis.org

3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 6, 6, 4, 5, 5, 6, 6, 6, 4, 5, 6, 6, 6, 7, 7, 4, 5, 7, 6, 7, 7, 7, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 4, 5, 6, 6, 7, 7, 8, 8, 8, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 8, 8, 4, 5, 7, 6, 7, 7, 8, 7, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 4, 5, 7, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 4, 5, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 4, 5, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 9
Offset: 1

Views

Author

Hugo Pfoertner, Jun 06 2017

Keywords

Comments

The table is given in the section "Results" of the notes by M. E. Pfetsch and G. M. Ziegler, see link.
An n X m convex lattice polygon presumably means an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - N. J. A. Sloane, Feb 07 2019

Examples

			Drawing the diagonals in a lattice square of size 1 X 1 produces 4 triangles, so T(1,1)=3.
Triangle begins:
  3;
  4, 4;
  4, 4, 4;
  4, 4, 5, 5;
  4, 5, 5, 6, 6;
  4, 5, 5, 6, 6, 6;
  4, 5, 6, 6, 6, 7, 7;
  ...

Links

Crossrefs

Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.

A288180 Number of intersection points formed by drawing the line segments connecting any two lattice points of an n X m convex lattice polygon written as triangle T(n,m), n >= 1, 1 <= m <= n.

Original entry on oeis.org

5, 13, 37, 35, 121, 353, 75, 265, 771, 1761, 159, 587, 1755, 4039, 8917, 275, 1019, 3075, 7035, 15419, 26773, 477, 1797, 5469, 12495, 27229, 47685, 84497, 755, 2823, 8693, 19831, 43333, 76357, 135075, 215545, 1163, 4369, 13301, 30333, 66699, 117719, 207643, 331233, 508613
Offset: 1

Views

Author

Hugo Pfoertner, Jun 06 2017

Keywords

Comments

If more than two lines intersect in the same point, only one intersection is counted.

Examples

			Triangle starts with:
n=1: 5,
n=2: 13, 37,
n=3: 35, 121, 353,
n=4: 75, 265, 771, 1761,
n=5: 159, 587, 1755, 4039, 8917,
n=6: 275, 1019, 3075, 7035, 15419, 26773,
n=7: 477, 1797, 5469, 12495, 27229, 47685, 84497,
n=8: 755, 2823, 8693, 19831, 43333, 76357, 135075, 215545,
n=9: 1163, 4369, 13301, 30333, 66699, 117719, 207643, 331233, 508613,
...

References

  • For references and links see A288177.

Links

Crossrefs

Cf. A288177, A288187.
For column 2 see A333279, A333280, A333281.
The main diagonal T(n,n) is A343993.

Extensions

Corrected and extended by Hugo Pfoertner, Jul 20 2017

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
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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