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 1-6 of 6 results.

A333276 a(n) = Sum_k k*A333274(n,k).

Original entry on oeis.org

12, 50, 152, 346, 732, 1294, 2232, 3546, 5428, 7806, 11136, 15226, 20676, 27150, 35048, 44386, 56044, 69302, 85480, 103882, 125180, 148942, 176968, 208034, 243772, 283014, 327272, 375826, 431212, 490918, 558456, 631978, 712844, 799726, 895152, 997322, 1110628
Offset: 1

Views

Author

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

Keywords

Comments

a(n)/A331755(n) is the average number of polygons touching a vertex in the graph defined in A306302.

Links

Crossrefs

Cf. A159065, A306302, A333274, A333275, A333277.

Extensions

a(15) and beyond from Lars Blomberg, Jun 17 2020

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.

A331755 Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

Original entry on oeis.org

2, 5, 13, 35, 75, 159, 275, 477, 755, 1163, 1659, 2373, 3243, 4429, 5799, 7489, 9467, 11981, 14791, 18275, 22215, 26815, 31847, 37861, 44499, 52213, 60543, 70011, 80347, 92263, 105003, 119557, 135327, 152773, 171275, 191721, 213547, 237953
Offset: 1

Views

Author

N. J. A. Sloane, Feb 02 2020

Keywords

Links

Crossrefs

Cf. A290131 (regions), A290132 (edges), A333274 (polygons per vertex), A333276, A159065.
For K_n see A007569, A007678, A135563.

Programs

  • Maple
    # Maple code from N. J. A. Sloane, Jul 16 2020
V106i := proc(n) local ans,a,b; ans:=0;
for a from 1 to n-1 do for b from 1 to n-1 do
if igcd(a,b)=1 then ans:=ans + (n-a)*(n-b); fi; od: od: ans; end; # A115004
V106ii := proc(n) local ans,a,b; ans:=0;
for a from 1 to floor(n/2) do for b from 1 to floor(n/2) do
if igcd(a,b)=1 then ans:=ans + (n-2*a)*(n-2*b); fi; od: od: ans; end; # A331761
A331755 := n -> 2*(n+1) + V106i(n+1) - V106ii(n+1);
  • Mathematica
    a[n_]:=Module[{x,y,s1=0,s2=0}, For[x=1, x<=n-1, x++, For[y=1, y<=n-1, y++, If[GCD[x,y]==1,s1+=(n-x)*(n-y); If[2*x<=n-1&&2*y<=n-1,s2+=(n-2*x)*(n-2*y)]]]]; s1-s2]; Table[a[n]+ 2 n, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2020 *)

Formula

a(n) = A290132(n) - A290131(n) + 1.
a(n) = A159065(n) + 2*n.
This is column 1 of A331453.
a(n) = (9/(8*Pi^2))*n^4 + O(n^3 log(n)). Asymptotic to (9/(2*Pi^2))*A000537(n-1). [Stéphane Legendre, see A159065.]

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

A334701 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly two lines cross.

Original entry on oeis.org

1, 6, 24, 54, 124, 214, 382, 598, 950, 1334, 1912, 2622, 3624, 4690, 6096, 7686, 9764, 12010, 14866, 18026, 21904, 25918, 30818, 36246, 42654, 49246, 57006, 65334, 75098, 85414, 97384, 110138, 124726, 139642, 156286, 174018, 194106, 214570, 237534, 261666, 288686, 316770, 348048, 380798, 416524, 452794, 492830
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, May 30 2020

Keywords

Comments

It would be nice to have a formula or recurrence. - N. J. A. Sloane, Jun 22 2020

Links

Crossrefs

Column 4 of array in A333275.
Cf. A306302, A331755, A290131, A333274.
See also A115004, A331761.

Formula

Conjecture: As n -> oo, a(n) ~ C*n^4/Pi^2, where C is about 0.95 (compare A115004, A331761). - N. J. A. Sloane, Jul 03 2020

Extensions

More terms from Lars Blomberg, Jun 17 2020

A333277 a(n) = Sum_k k*A333275(n,k).

Original entry on oeis.org

4, 30, 116, 290, 652, 1186, 2092, 3370, 5212, 7546, 10828, 14866, 20260, 26674, 34508, 43778, 55364, 68546, 84644, 102962, 124172, 147842, 175772, 206738, 242372, 281506, 325652, 374090, 429356, 488938, 556348, 629738, 710468, 797210, 892492, 994514, 1107668
Offset: 1

Views

Author

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

Keywords

Comments

a(n)/A331755(n) is the average number of polygons touching a non-boundary vertex in the graph defined in A306302.

Links

Crossrefs

Cf. A159065, A306302, A333274, A333275, A333276.

Extensions

a(6) and beyond from Lars Blomberg, Jun 17 2020
Showing 1-6 of 6 results.

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

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