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 1-4 of 4 results.

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.]

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

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 1-4 of 4 results.

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

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