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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A247268 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Y, U, X.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 0, 4, 5, 38, 22, 13, 90, 144, 457, 408, 386, 1267, 2230, 5912, 6481, 7098, 18896, 35433, 79634, 101232, 127501, 288304, 546652, 1113907, 1560356, 2148298, 4408181, 8335234, 15954116, 23827541, 35011426, 67591204, 126376945, 232719926
Offset: 0

Views

Author

Alois P. Heinz, Nov 30 2014

Keywords

Examples

			a(3) = 1, a(5) = 2:
._____.     ._________.   ._________.
| ._. |     |_. .___| |   | |___. ._|
|_| |_|     | |_| |_. |   | ._| |_| |
|_. ._|  ,  | |_. ._| |   | |_. ._| |
| |_| |     | ._|_| |_|   |_| |_|_. |
|_____|     |_|_______|   |_______|_|  .

Links

Crossrefs

Cf. A174249, A233427, A247124, A247125.

Programs

  • Maple
    gf:= -(x^40 +12*x^39 +36*x^38 -5*x^36 -2*x^35 +12*x^34 +54*x^33 +4*x^32 -21*x^31 -23*x^30 +4*x^29 +20*x^28 +4*x^27 -4*x^25 -7*x^24 -6*x^23 -3*x^22 +33*x^21 -7*x^20 -10*x^19 -12*x^18 -9*x^17 +12*x^16 +16*x^15 +3*x^14 -2*x^13 -2*x^12 -2*x^11 -3*x^10 +5*x^9 -2*x^6 -7*x^5 -x^4 +1) /
(x^43 +12*x^42 +36*x^41 -3*x^40 -29*x^39 -58*x^38 +12*x^37 +67*x^36 +4*x^35 -123*x^34 -99*x^33 +8*x^32 +23*x^31 -145*x^30 -52*x^29 -52*x^28 -35*x^27 -112*x^26 -99*x^25 -28*x^24 -7*x^23 -15*x^22 -99*x^21 -42*x^20 +22*x^19 +36*x^18 +26*x^17 -4*x^16 +6*x^15 +31*x^14 +5*x^13 +11*x^12 +14*x^11 +23*x^10 -5*x^9 -7*x^8 -x^7 +2*x^6 +9*x^5 +x^4 +x^3 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

Formula

G.f.: see Maple program.

A247706 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape P; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 3, 0, 2, 16, 20, 20, 0, 135, 204, 140, 16, 6, 944, 1432, 1164, 296, 170, 0, 4814, 8796, 8452, 4068, 1708, 92, 20, 26435, 58656, 66994, 41648, 17494, 2700, 762, 0, 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62, 978044, 2783560, 3836254, 3107308, 1696312, 609772, 172724, 18220, 3160, 0
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247739(n).

Examples

			T(2,2) = 2:
.___.   .___.
|   |   |   |
| ._|   |_. |
|_| |   | |_|
|   |   |   |
|___|   |___| .
Triangle T(n,k) begins:
00 :      1;
01 :      1,      0;
02 :      3,      0,      2;
03 :     16,     20,     20,      0;
04 :    135,    204,    140,     16,      6;
05 :    944,   1432,   1164,    296,    170,     0;
06 :   4814,   8796,   8452,   4068,   1708,    92,    20;
07 :  26435,  58656,  66994,  41648,  17494,  2700,   762,   0;
08 : 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247770.
Even bisection of main diagonal gives A247076.
Cf. A247739.

A247711 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape X; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 5, 55, 1, 493, 8, 3930, 76, 27207, 734, 9, 207118, 7414, 157, 1622723, 71986, 2064, 8, 12544364, 638499, 22232, 259, 95912510, 5558790, 222964, 3898, 50, 732066083, 47971603, 2179607, 49537, 948, 8, 5616480627, 410502410, 20604626, 564498, 13889, 180
Offset: 0

Views

Author

Alois P. Heinz, Sep 23 2014

Keywords

Comments

Sum_{k>0} k * T(n,k) = A247744(n).

Examples

			T(3,1) = 1:
._____.
| ._. |
|_| |_|
|_. ._|
| |_| |
|_____|
.
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       55,       1;
04 :      493,       8;
05 :     3930,      76;
06 :    27207,     734,      9;
07 :   207118,    7414,    157;
08 :  1622723,   71986,   2064,    8;
09 : 12544364,  638499,  22232,  259;
10 : 95912510, 5558790, 222964, 3898, 50;

Links

Crossrefs

Row sums give A174249 or A233427(n,5).
Columns k=0-1 give: A247775, A247828.
Cf. A247744.

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

Original entry on oeis.org

1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049
Offset: 0

Views

Author

Alois P. Heinz, Nov 25 2015

Keywords

Examples

			a(4) = 13:
._______.      ._______.      ._______.      ._______.
| | | | |      |   |   |      |   | | |      |   ._| |
| | | | |      | ._| ._|      | ._| | |      |___|   |
| | | | |      |_| |_| |      |_| | | |      |   |___|
| | | | | (1)  |   |   | (4)  |   | | | (6)  | ._|   | (2)
|_|_|_|_|      |___|___|      |_ _|_|_|      |_|_____|    .
a(5) = 52:
._________.
|   |_.   |
| ._| |___|
|_|_   _| |
|   |_|   | (2)  ...
|_____|___|          .

Links

Crossrefs

Cf. A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A278330.

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Original entry on oeis.org

1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372
Offset: 0

Views

Author

Alois P. Heinz, Nov 18 2016

Keywords

Examples

			a(2) = 2,          a(3) = 1:
.___.   .___.      ._____.
|   |   |   |      | ._. |
| ._|   |_. |      |_| |_|
|_| |   | |_|      |_   _|
|   |   |   |      | |_| |
|___|   |___|      |_____| .

Links

Crossrefs

Cf. A079978, A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A264812.

Programs

  • Maple
    a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
    <<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);

Formula

G.f.: -(4*x^6+x^3-1) / (8*x^12 +16*x^11 +6*x^9 +4*x^8 +8*x^7 -21*x^6 -4*x^5 -8*x^4 -2*x^3 -2*x^2+1).
a(n) mod 2 = A079978(n).

A278456 Number of tilings of a 5 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 2, 50, 1954, 56864, 1532496, 42238426, 1178422563, 32890293494, 917103556607, 25552076570350, 711923354658732, 19838824712825618, 552851181380560869, 15406086995815163663, 429312063890812931103, 11963383230714027535776, 333377000620725693771782
Offset: 0

Views

Author

Alois P. Heinz, Nov 22 2016

Keywords

Examples

			a(1) = 2:
._.   ._.
|_|   | |
|_|   | |
|_|   | |
|_|   | |
|_|   |_|  .

Links

Crossrefs

Column k=5 of A278657.
Cf. A174249, A233427, A273477.

A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

Original entry on oeis.org

1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
.___. ._____. ._____.
| .. | | .. | | | | ._. |
|| || || || | | || ||
|. .| , | .| .| |. |. |
| || | | | || | | |_| | |
|___| ||____| |___|_|. - Alois P. Heinz, Jan 03 2014

Links

Crossrefs

Partial sums of A077976.
Cf. A174249, A233427, A234312, A234931, A247126.

Programs

  • Maple
    a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
        <<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 20 2013
  • Mathematica
    CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)
  • PARI
    Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013

Formula

a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013
G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013

A174250 Number of tilings of a 6 X n rectangle with n hexominoes of any shape.

Original entry on oeis.org

1, 1, 6, 132, 2369, 33344, 451206, 5850115, 81459922, 1144259389, 15946621499
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A134438, A174248, A174249, A174251, A174252, A174253.

A174251 Number of tilings of a 7 X n rectangle with n heptominoes of any shape.

Original entry on oeis.org

1, 1, 7, 259, 9525, 270827, 6633399, 158753814, 3825111851, 96608374284, 2446223788303
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A134438, A174248, A174249, A174250, A174252, A174253.

A174252 Number of tilings of an 8 X n rectangle with n octominoes of any shape.

Original entry on oeis.org

1, 1, 8, 546, 39731, 2152050, 99697633, 4292655082, 187497290034, 8378760802160, 385296986628990
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A134438, A174248, A174249, A174250, A174251, A174253.
Previous Showing 11-20 of 81 results. Next

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

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