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 1-7 of 7 results.

A302336 Linear coefficient (in absolute value) of the quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1.

Original entry on oeis.org

0, 2, 6, 28, 140, 740, 4056, 22904, 132344, 778832, 4652404, 28140536, 172021360, 1061153560, 6597813620, 41307119692, 260198053200, 1647958588568, 10488324116052, 67046234983840, 430300354820176, 2771678138269600, 17912347088664868, 116113406138798112
Offset: 1

Views

Author

Eric W. Weisstein, Apr 05 2018

Keywords

Examples

			Let p(k,n) be the number of 2k-cycles in the n X n grid graph for n >= k-1. p(k,n) are quadratic polynomials in n, with the first few given by:
  p(1,n) =      0,
  p(2,n) =      1 -      2*n +       n^2,
  p(3,n) =      4 -      6*n +     2*n^2,
  p(4,n) =     26 -     28*n +     7*n^2,
  p(5,n) =    164 -    140*n +    28*n^2,
  p(6,n) =   1046 -    740*n +   124*n^2,
  p(7,n) =   6672 -   4056*n +   588*n^2,
  p(8,n) =  42790 -  22904*n +  2938*n^2,
  p(9,n) = 275888 - 132344*n + 15268*n^2,
  ...
The linear coefficients give a(n), so the first few are 0, 2, 6, 28, 140, .... - _Eric W. Weisstein_, Apr 05 2018

Links

Crossrefs

Cf. A302335 (constant coefficients).
Cf. A002931 (quadratic coefficients).
Cf. A006772, A302337.

Formula

a(n) = 2*A006772(n). - Andrey Zabolotskiy, Nov 09 2018

Extensions

Terms a(12) and beyond added using data from A006772 by Andrey Zabolotskiy, Feb 10 2022

A333651 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2, read by rows, where T(n,k) is the number of 2*(k+2)-cycles in the n X n grid graph which pass through NW corner (0,0).

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 6, 18, 40, 24, 6, 1, 2, 6, 20, 72, 248, 698, 1100, 1096, 662, 206, 1, 2, 6, 20, 74, 298, 1228, 4762, 15984, 40026, 75524, 109150, 121130, 99032, 51964, 11996, 1072, 1, 2, 6, 20, 74, 300, 1300, 5844, 26148, 110942, 427388, 1393796, 3790524, 8648638, 16727776, 27529284, 38120312, 43012614, 37385280, 23166526, 9496426, 2286972, 242764
Offset: 2

Views

Author

Seiichi Manyama, Apr 01 2020

Keywords

Examples

			T(3,0) = 1;
   +--*
   |  |
   *--*
T(3,1) = 2;
   +--*--*   +--*
   |     |   |  |
   *--*--*   *  *
             |  |
             *--*
T(3,2) = 4;
   +--*--*   +--*--*   +--*--*   +--*
   |     |   |     |   |     |   |  |
   *     *   *  *--*   *--*  *   *  *--*
   |     |   |  |         |  |   |     |
   *--*--*   *--*         *--*   *--*--*
Triangle starts:
===================================================
n\k| 0  1  2   3   4    5     6 ...     10 ...  16
---|-----------------------------------------------
2  | 1;
3  | 1, 2, 4;
4  | 1, 2, 6, 18, 40,  24,    6;
5  | 1, 2, 6, 20, 72, 248,  698, ... , 206;
6  | 1, 2, 6, 20, 74, 298, 1228, .......... , 1072;
7  | 1, 2, 6, 20, 74, 300, 1300, ...
8  | 1, 2, 6, 20, 74, 300, 1302, ...
9  | 1, 2, 6, 20, 74, 300, 1302, ...

Links

Crossrefs

Row sums give A333246.
Cf. A003763, A034010, A302337, A333652, A333667, A333668.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333651(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1)
    return [cycles.len(2 * k).len() for k in range(2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333651(n)])

Formula

T(n,k) = A034010(k+2) for k <= n-2.

A333652 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-n, read by rows, where T(n,k) is the number of 2*(k+n)-cycles in the n X n grid graph which pass through NW and SW corners.

Original entry on oeis.org

1, 1, 3, 1, 6, 17, 17, 6, 1, 10, 45, 167, 404, 570, 460, 186, 1, 15, 100, 506, 2164, 7726, 20483, 39401, 56015, 57632, 37450, 10340, 1072, 1, 21, 196, 1316, 7066, 33983, 147377, 546400, 1656592, 4099732, 8394433, 14227675, 19443270, 20239262, 14767415, 7007270, 1926990, 230440
Offset: 2

Views

Author

Seiichi Manyama, Apr 01 2020

Keywords

Examples

			T(3,0) = 1;
   +--*
   |  |
   *  *
   |  |
   +--*
T(3,1) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *     *   *  *--*   *  *--*
   |     |   |  |      |     |
   +--*--*   +--*      +--*--*
Triangle starts:
====================================================================
n\k| 0   1    2     3      4 ...      7 ...  12 ...    17 ...    24
---|----------------------------------------------------------------
2  | 1;
3  | 1,  3;
4  | 1,  6,  17,   17,     6;
5  | 1, 10,  45,  167,   404, ... , 186;
6  | 1, 15, 100,  506,  2164, .......... , 1072;
7  | 1, 21, 196, 1316,  7066, .................. , 230440;
8  | 1, 28, 350, 3038, 20317, ............................ , 4638576;

Links

Crossrefs

Row sums give A333247.
Cf. A000217, A003763, A302337, A333651, A333667, A333668.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333652(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n)
    return [cycles.len(2 * k).len() for k in range(n, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333652(n)])

Formula

T(n,0) = 1.
T(n,1) = A000217(n-1) for n > 2.

A333667 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).

Original entry on oeis.org

1, 3, 20, 16, 6, 175, 420, 562, 456, 186, 1764, 8064, 21224, 39500, 55376, 57248, 37586, 10260, 1072, 19404, 138600, 569768, 1717152, 4151965, 8371428, 14126846, 19364732, 20241450, 14759356, 6998166, 1927724, 230440
Offset: 2

Views

Author

Seiichi Manyama, Apr 01 2020

Keywords

Examples

			T(3,0) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *--*  *   *     *   *  *--*
      |  |   |     |   |     |
      *--+   *--*--+   *--*--+
Triangle starts:
=======================================================================
n\k|      0        1         2 ...      4 ...   8 ...    12 ...     18
---|-------------------------------------------------------------------
2  |      1;
3  |      3;
4  |     20,      16,        6;
5  |    175,     420,      562, ... , 186;
6  |   1764,    8064,    21224, .......... , 1072;
7  |  19404,  138600,   569768, .................. , 230440;
8  | 226512, 2265120, 12922446, ............................ , 4638576;

Links

Crossrefs

Row sums give A333323.
Cf. A003763, A302337, A333651, A333652, A333668.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333667(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n * n)
    return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333667(n)])

Formula

T(n,0) = A000891(n-2).

A333668 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

Original entry on oeis.org

1, 1, 1, 4, 6, 1, 12, 58, 156, 146, 1, 24, 244, 1416, 5435, 12976, 16654, 7108, 1072, 1, 40, 696, 7076, 47965, 236628, 873610, 2348664, 4335724, 4958224, 3407276, 1298704, 205792
Offset: 2

Views

Author

Seiichi Manyama, Apr 01 2020

Keywords

Examples

			T(4,1) = 4;
   +--*--*--+   +--*--*--+   +--*--*--+   +--*  *--+
   |        |   |        |   |        |   |  |  |  |
   *--*     *   *     *--*   *        *   *  *--*  *
      |     |   |     |      |        |   |        |
   *--*     *   *     *--*   *  *--*  *   *        *
   |        |   |        |   |  |  |  |   |        |
   +--*--*--+   +--*--*--+   +--*  *--+   +--*--*--+
Triangle starts:
=================================================================
n\k| 0   1     2      3       4 ...       8 ...    12 ...     18
---|-------------------------------------------------------------
2  | 1;
3  | 1;
4  | 1,  4,    6;
5  | 1, 12,   58,   156,    146;
6  | 1, 24,  244,  1416,   5435, ... , 1072;
7  | 1, 40,  696,  7076,  47965, ........... , 205792;
8  | 1, 60, 1590, 24960, 263770, ..................... , 4638576;

Links

Crossrefs

Row sums give A333466.
Cf. A003763, A046092, A302337, A333651, A333652, A333667.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333668(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, n, n * (n - 1) + 1, n * n]:
        cycles = cycles.including(i)
    return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333668(n)])

Formula

T(n,0) = 1.
T(n,1) = A046092(n-3).

A006772 Sum of spans of 2n-step polygons on square lattice.

Original entry on oeis.org

0, 1, 3, 14, 70, 370, 2028, 11452, 66172, 389416, 2326202, 14070268, 86010680, 530576780, 3298906810, 20653559846, 130099026600, 823979294284, 5244162058026, 33523117491920, 215150177410088, 1385839069134800, 8956173544332434, 58056703069399056, 377396656568011618, 2459614847765495754, 16068572108927106202
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Andrey Zabolotskiy_, Nov 09 2018: (Start)
There are no 2-step polygons (conventionally).
For n=2, the only 4-step polygon is a 1 X 1 square having span 1, so a(2)=1.
For n=3, the only 6-step polygon is a 2 X 1 domino which can be rotated 2 ways having spans 2 and 1, so a(3) = 2+1 = 3.
For n=4, there are the following 8-step polygons:
a 3 X 1 stick which can be rotated 2 ways having spans 3 and 1;
an L-tromino which can be rotated 4 ways, all having span 2;
a 2 X 2 square, having span 2.
So a(4) = 3 + 1 + 4*2 + 2 = 14.
For n=5, there are the following 10-step polygons:
a 4 X 1 stick which can be rotated 2 ways having spans 4 and 1;
an L-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3, plus reflections;
a T-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3;
an S-tetromino which can be rotated 2 ways having spans 3 and 2, plus reflections;
a 3 X 2 rectangle which can be rotated 2 ways having spans 3 and 2;
a 3 X 2 rectangle without one of its angular squares having same counts as L-tetromino.
So a(5) = 4 + 1 + 2 * 2*2*(2+3) + 2*(2+3) + 2*(3+2) + 3 + 2 = 70.
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002931, A006773, A302336, A302337, A232103.

Extensions

Name corrected, more terms from Andrey Zabolotskiy, Nov 09 2018

A333520 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length 2*(n-1+k) connecting opposite corners in the n X n grid graph (0 <= k <= floor((n-1)^2/2), n >= 1).

Original entry on oeis.org

1, 2, 6, 4, 2, 20, 36, 48, 48, 32, 70, 224, 510, 956, 1586, 2224, 2106, 732, 104, 252, 1200, 3904, 10560, 25828, 58712, 121868, 217436, 300380, 280776, 170384, 61336, 10180, 924, 5940, 25186, 88084, 277706, 821480, 2309402, 6140040, 15130410, 33339900, 62692432, 96096244, 116826664, 110195700, 78154858, 39287872, 12396758, 1879252, 111712
Offset: 1

Views

Author

Seiichi Manyama, Mar 29 2020

Keywords

Examples

			T(3,1) = 4;
   S--*      S--*--*   S  *--*   S
      |            |   |  |  |   |
   *--*         *--*   *--*  *   *  *--*
   |            |            |   |  |  |
   *--*--E      *--E         E   *--*  E
Triangle starts:
=======================================================
n\k|   0     1     2      3      4 ...      8 ...   12
---|---------------------------------------------------
1  |   1;
2  |   2;
3  |   6,    4,    2;
4  |  20,   36,   48,    48,    32;
5  |  70,  224,  510,   956,  1586, ... , 104;
6  | 252, 1200, 3904, 10560, ................. , 10180;

Links

Crossrefs

Row sums give A007764.
T(n,0) gives A000984(n-1).
T(n,1) gives A257888(n).
T(n,floor((n-1)^2/2)) gives A121788(n-1).
T(2*n-1,2*(n-1)^2) gives A001184(n-1).
Cf. A074148, A302337, A329633.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333520(n):
    if n == 1: return [1]
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return [paths.len(2 * (n - 1 + k)).len() for k in range((n - 1) ** 2 // 2 + 1)]
print([i for n in range(1, 8) for i in A333520(n)])
Showing 1-7 of 7 results.

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

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