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.

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

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.

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

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, May 23 2020

Keywords

Examples

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

Links

Crossrefs

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

Formula

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

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

A334694 a(n) = (n/4)*(n^3+2*n^2+5*n+8).

Original entry on oeis.org

0, 4, 17, 51, 124, 260, 489, 847, 1376, 2124, 3145, 4499, 6252, 8476, 11249, 14655, 18784, 23732, 29601, 36499, 44540, 53844, 64537, 76751, 90624, 106300, 123929, 143667, 165676, 190124, 217185, 247039, 279872, 315876, 355249, 398195, 444924, 495652, 550601, 609999, 674080, 743084, 817257, 896851, 982124, 1073340
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, May 19 2020

Keywords

Comments

Consider a figure made up of a row of n >= 1 adjacent congruent rectangles in which all possible diagonals of the rectangles have been drawn. The number of regions formed is A306302. If we distort all these diagonals very slightly so that no three lines meet at a point, the number of regions changes to a(n).

Links

Crossrefs

Cf. A306302, A331452, A333275.

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{0,4,17,51,124},50] (* or *)
A334694[n_]:=n/4(n^3+2n^2+5n+8);Array[A334694,50,0] (* Paolo Xausa, Nov 08 2023 *)
  • PARI
    concat(0, Vec(x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, May 27 2020

Formula

Satisfies the identity a(n) = A306302(n) + Sum_{k=3..(n+1)} binomial(k-1,2)*A333275(n,2*k). E.g. for n=4 we have a(4) = 104 + 8*1 + 2*3 + 1*6 = 124.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)
Showing 1-6 of 6 results.

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

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