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.

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A174253 Number of tilings of a 9 X n rectangle with n nonominoes of any shape.

Original entry on oeis.org

1, 1, 9, 1095, 145415, 15661597, 1418159011, 112976454947, 8812020831683, 706152947468301, 58015563977931125
Offset: 0

Views

Author

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

Keywords

Links

Crossrefs

Cf. A134438, A174248, A174249, A174250, A174251, A174252.

A247124 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, U, X.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 14, 21, 37, 63, 122, 221, 374, 656, 1147, 2066, 3699, 6477, 11407, 20099, 35656, 63323, 111775, 197352, 348556, 616560, 1091570, 1929721, 3410509, 6028021, 10658114, 18851012, 33331681, 58927069, 104177155, 184188343, 325686763, 575858676
Offset: 0

Views

Author

Alois P. Heinz, Nov 19 2014

Keywords

Examples

			a(4) = 3:
._______.   ._______.   ._______.
| | | | |   | | ._. |   | ._. | |
| | | | |   | |_| |_|   |_| |_| |
| | | | |   | |_. ._|   |_. ._| |
| | | | |   | | |_| |   | |_| | |
|_|_|_|_|   |_|_____|   |_____|_|  .

Links

Crossrefs

Cf. A174249, A233427, A247125, A247268, A264812.

Programs

  • Maple
    gf:= -(x-1)^2 *(x^4+x^3+x^2+x+1)^2 /
     (x^15 +x^13 +x^11 -3*x^10 -2*x^8 -2*x^6 +6*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

Formula

G.f.: see Maple program.

A247702 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape F; triangle T(n,k), n>=0, 0<=k<=max(delta_{3,n},floor((n-2)/2)*2), read by rows.

Original entry on oeis.org

1, 1, 5, 52, 4, 437, 60, 4, 3342, 584, 80, 21734, 5372, 818, 24, 2, 155685, 49540, 8800, 620, 44, 1153475, 439780, 92500, 10140, 856, 28, 2, 8422634, 3726836, 914142, 127596, 13338, 760, 48, 60853524, 30683256, 8544440, 1425320, 176156, 14404, 1078, 32, 2
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247735(n).

Examples

			T(3,1) = 4:
._____.  ._____.  ._____.  ._____.
|_.   |  |   ._|  | ._. |  | ._. |
| |___|  |___| |  |_| |_|  |_| |_|
|_. ._|  |_. ._|  | .___|  |___. |
| |_| |  | |_| |  |_|   |  |   |_|
|_____|  |_____|  |_____|  |_____| .
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       52,        4;
04 :      437,       60,       4;
05 :     3342,      584,      80;
06 :    21734,     5372,     818,      24,      2;
07 :   155685,    49540,    8800,     620,     44;
08 :  1153475,   439780,   92500,   10140,    856,    28,    2;
09 :  8422634,  3726836,  914142,  127596,  13338,   760,   48;
10 : 60853524, 30683256, 8544440, 1425320, 176156, 14404, 1078, 32, 2;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247766.
Cf. A247735.

A247703 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape I; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 4, 0, 1, 47, 8, 0, 1, 394, 94, 12, 0, 1, 2082, 1608, 282, 32, 0, 2, 15113, 8812, 3452, 512, 58, 0, 3, 111664, 73863, 22310, 5962, 790, 96, 0, 4, 789930, 631700, 218608, 45762, 9374, 1260, 142, 0, 5, 5388729, 5157928, 2067811, 491868, 81720, 15272, 1824, 196, 0, 6
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247736(n).

Examples

			T(5,5) = 2:
._._._._._.   ._________.
| | | | | |   |_________|
| | | | | |   |_________|
| | | | | |   |_________|
| | | | | |   |_________|
|_|_|_|_|_|   |_________| .
Triangle T(n,k) begins:
00 :      1;
01 :      0,      1;
02 :      4,      0,      1;
03 :     47,      8,      0,     1;
04 :    394,     94,     12,     0,    1;
05 :   2082,   1608,    282,    32,    0,    2;
06 :  15113,   8812,   3452,   512,   58,    0,   3;
07 : 111664,  73863,  22310,  5962,  790,   96,   0,  4;
08 : 789930, 631700, 218608, 45762, 9374, 1260, 142,  0,  5;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247767.
Main diagonal gives A003520.
Cf. A247736.

A247704 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape L; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 3, 0, 2, 36, 16, 4, 0, 177, 220, 100, 0, 4, 1300, 1720, 816, 144, 26, 0, 8866, 11152, 5616, 1784, 524, 0, 8, 54849, 85016, 51116, 18380, 4656, 584, 88, 0, 372943, 622732, 448744, 189360, 52130, 8948, 1908, 0, 16, 2466986, 4528336, 3670116, 1806160, 582250, 127140, 22206, 1912, 248, 0
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247737(n).

Examples

			T(2,2) = 2:
.___.   .___.
| ._|   |_. |
| | |   | | |
| | |   | | |
|_| |   | |_|
|___|   |___| .
Triangle T(n,k) begins:
00 :      1;
01 :      1,      0;
02 :      3,      0,      2;
03 :     36,     16,      4,      0;
04 :    177,    220,    100,      0,     4;
05 :   1300,   1720,    816,    144,    26,    0;
06 :   8866,  11152,   5616,   1784,   524,    0,    8;
07 :  54849,  85016,  51116,  18380,  4656,  584,   88,  0;
08 : 372943, 622732, 448744, 189360, 52130, 8948, 1908,  0, 16;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247768.
Cf. A247737.

A247705 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape N; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 5, 48, 8, 423, 68, 10, 3082, 832, 84, 8, 18998, 7624, 1230, 88, 10, 133083, 65360, 14390, 1732, 116, 8, 965175, 555236, 150876, 23184, 2196, 108, 6, 6907447, 4531744, 1454292, 275320, 33807, 2616, 124, 4, 48357538, 36466396, 13354738, 3012116, 457360, 46872, 3086, 104, 2
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247738(n).

Examples

			T(3,1) = 8:
._____.        .___._.
| ._. |        | ._| |
|_| |_|        | | ._|
| ._| |        | | | |
| |   |        |_|_| |
|_|___| (*4)   |_____| (*4)  .
Triangle T(n,k) begins:
00 :      1;
01 :      1;
02 :      5;
03 :     48,      8;
04 :    423,     68,     10;
05 :   3082,    832,     84,     8;
06 :  18998,   7624,   1230,    88,   10;
07 : 133083,  65360,  14390,  1732,  116,   8;
08 : 965175, 555236, 150876, 23184, 2196, 108,  6;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247769.
Cf. A247738.

A247707 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape T; triangle T(n,k), n>=0, 0<=k<=max(0,floor(2*(n-1)/3)), read by rows.

Original entry on oeis.org

1, 1, 5, 50, 6, 437, 62, 2, 3270, 700, 36, 21720, 5712, 506, 12, 160593, 48364, 5444, 282, 6, 1209537, 425638, 57648, 3836, 122, 8999307, 3578302, 576791, 48688, 2226, 40, 66054288, 29550476, 5500946, 558036, 33400, 1056, 10, 485082083, 239927980, 50762537, 6035146, 440480, 19180, 380
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247740(n).

Examples

			T(4,2) = 2:
._____._.    ._._____.
|_. ._| |    | |_. ._|
| | |_. |    | ._| | |
| |_| | |    | | |_| |
| ._| |_|    |_| |_. |
|_|_____|    |_____|_| .
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       50,        6;
04 :      437,       62,       2;
05 :     3270,      700,      36;
06 :    21720,     5712,     506,     12;
07 :   160593,    48364,    5444,    282,     6;
08 :  1209537,   425638,   57648,   3836,   122;
09 :  8999307,  3578302,  576791,  48688,  2226,   40;
10 : 66054288, 29550476, 5500946, 558036, 33400, 1056, 10;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247771.
Cf. A247740.

A247708 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape U; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 5, 39, 16, 1, 369, 120, 12, 2908, 1000, 98, 19185, 7474, 1228, 60, 3, 137200, 63896, 12448, 1092, 53, 1022915, 540562, 120034, 12676, 590, 4, 7606043, 4365686, 1084022, 140512, 8836, 250, 5, 55699672, 34738058, 9663366, 1466724, 124242, 5984, 166
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247741(n).

Examples

			T(3,2) = 1:
._____.
| ._. |
|_| |_|
|_. ._|
| |_| |
|_____| .
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       39,       16,       1;
04 :      369,      120,      12;
05 :     2908,     1000,      98;
06 :    19185,     7474,    1228,      60,      3;
07 :   137200,    63896,   12448,    1092,     53;
08 :  1022915,   540562,  120034,   12676,    590,    4;
09 :  7606043,  4365686, 1084022,  140512,   8836,  250,   5;
10 : 55699672, 34738058, 9663366, 1466724, 124242, 5984, 166;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247772.
Cf. A247741.

A247709 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape V; triangle T(n,k), n>=0, 0<=k<=max(0,n-2+delta_{n,3}), read by rows.

Original entry on oeis.org

1, 1, 5, 38, 16, 2, 329, 152, 20, 2614, 1224, 160, 8, 17400, 8656, 1714, 168, 12, 122843, 72104, 17280, 2300, 158, 4, 901647, 598444, 168422, 25872, 2284, 108, 4, 6662758, 4770520, 1479850, 260672, 29166, 2256, 124, 8, 48492622, 37416964, 12800398, 2601524, 351578, 32840, 2182, 100, 4
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247742(n).

Examples

			T(3,2) = 2:
._____.    ._____.
| .___|    |___. |
| | ._|    |_. | |
|_| | |    | | |_|
|___| |    | |___|
|_____|    |_____| .
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       38,       16,        2;
04 :      329,      152,       20;
05 :     2614,     1224,      160,       8;
06 :    17400,     8656,     1714,     168,     12;
07 :   122843,    72104,    17280,    2300,    158,     4;
08 :   901647,   598444,   168422,   25872,   2284,   108,    4;
09 :  6662758,  4770520,  1479850,  260672,  29166,  2256,  124,   8;
10 : 48492622, 37416964, 12800398, 2601524, 351578, 32840, 2182, 100, 4;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247773.
Cf. A247742.

A247710 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape W; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/2)*2) read by rows.

Original entry on oeis.org

1, 1, 5, 56, 461, 32, 8, 3558, 368, 80, 23966, 3256, 696, 24, 8, 178127, 29564, 6558, 360, 80, 1362597, 266672, 61858, 4852, 770, 24, 8, 10194184, 2361632, 581452, 58732, 8890, 384, 80, 75684682, 20056764, 5220634, 632044, 97174, 5968, 914, 24, 8
Offset: 0

Views

Author

Alois P. Heinz, Sep 23 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247743(n).

Examples

			T(4,2) = 8:
._______.        ._______.        ._______.
| ._____|        |_. |_. |        | ._____|
|_| ._| |        | |_. | |        |_| ._| |
| ._| ._|        | | |_| |        | ._| | |
|_|___| |        | |_. |_|        |_| ._| |
|_______| (*2)   |___|___| (*2)   |___|___| (*4)
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       56;
04 :      461,       32,       8;
05 :     3558,      368,      80;
06 :    23966,     3256,     696,     24,     8;
07 :   178127,    29564,    6558,    360,    80;
08 :  1362597,   266672,   61858,   4852,   770,   24,   8;
09 : 10194184,  2361632,  581452,  58732,  8890,  384,  80;
10 : 75684682, 20056764, 5220634, 632044, 97174, 5968, 914, 24, 8;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247774.
Cf. A247743.
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