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 1-10 of 20 results. Next

A238009 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 3 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 3, 8, 3, 1, 3, 12, 8, 1, 4, 18, 22, 6, 1, 4, 24, 40, 22, 1, 5, 32, 73, 66, 10, 1, 5, 40, 112, 146, 48, 1, 6, 50, 172, 292, 174, 20, 1, 6, 60, 240, 516, 448, 116, 1, 7, 72, 335, 860, 1020, 464, 36, 1, 7, 84, 440, 1340, 2016, 1360, 256
Offset: 2

Views

Author

Christopher Hunt Gribble, Feb 16 2014

Keywords

Examples

			The first 19 rows of T(n,k) are:
   n\k 0  1   2    3    4     5     6     7     8    9  10
   2   1  1
   3   1  1
   4   1  2   2
   5   1  2   4
   6   1  3   8    3
   7   1  3  12    8
   8   1  4  18   22    6
   9   1  4  24   40   22
  10   1  5  32   73   66    10
  11   1  5  40  112  146    48
  12   1  6  50  172  292   174    20
  13   1  6  60  240  516   448   116
  14   1  7  72  335  860  1020   464    36
  15   1  7  84  440 1340  2016  1360   256
  16   1  8  98  578 2010  3716  3400  1168    72
  17   1  8 112  728 2890  6336  7432  3840   584
  18   1  9 128  917 4046 10326 14864 10600  2920  136
  19   1  9 144 1120 5502 16016 27536 25344 10600 1280
  20   1 10 162 1368 7336 24066 48188 54992 31800 7080 272

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Programs

  • PARI
    T(n,k)={(2^k*binomial(n-1*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-1*k-(k%2)-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/2), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Corrected C++ program and xrefs added by Christopher Hunt Gribble, Apr 25 2015

A231145 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=n-n%2, read by rows.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 1, 4, 13, 10, 4, 1, 4, 23, 35, 23, 1, 6, 40, 101, 125, 54, 10, 1, 6, 58, 206, 403, 336, 106, 1, 8, 83, 392, 1056, 1438, 956, 240, 25, 1, 8, 109, 641, 2281, 4424, 4718, 2409, 473, 1, 10, 142, 1011, 4429, 11370, 17252, 14478, 6094, 1020, 70
Offset: 2

Views

Author

Christopher Hunt Gribble, Feb 23 2014

Keywords

Examples

			The first 8 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8
n
2     1     2     2
3     1     2     4
4     1     4    13    10     4
5     1     4    23    35    23
6     1     6    40   101   125    54    10
7     1     6    58   206   403   336   106
8     1     8    83   392  1056  1438   956   240    25
9     1     8   109   641  2281  4424  4718  2409   473

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

A231473 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 4, 4, 1, 6, 9, 1, 6, 18, 1, 8, 28, 10, 1, 8, 42, 28, 1, 10, 57, 76, 1, 10, 76, 140, 25, 1, 12, 96, 254, 107, 1, 12, 120, 392, 321, 1, 14, 145, 600, 731, 70, 1, 14, 174, 840, 1462, 366, 1, 16, 204, 1170, 2610, 1308, 1, 16, 238, 1540, 4350, 3416, 196
Offset: 3

Views

Author

Christopher Hunt Gribble, Feb 23 2014

Keywords

Examples

			The first 11 rows of T(n,k) are:
.\ k    0      1      2      3      4
n
3       1      2
4       1      2
5       1      4
6       1      4      4
7       1      6      9
8       1      6     18
9       1      8     28     10
10      1      8     42     28
11      1     10     57     76
12      1     10     76    140     25
13      1     12     96    254    107

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Programs

  • Mathematica
    T[n_, k_] := ((3^k + 1)*Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]]*(3^(Quotient[(k + 1), 2]) + 3^Quotient[k, 2]) Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
  • PARI
    T(n,k)={((3^k+1)*binomial(n-2*k,k) + (k%2==0||n%2==1) * (3^((k+1)\2)+3^(k\2)) * binomial((n-2*k-(n%2))/2,k\2))/4}
for(n=3,20,for(k=0,floor(n/3), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015
Terms a(40) and beyond from Andrew Howroyd, May 29 2017

A232440 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 4, 1, 5, 6, 1, 5, 9, 1, 6, 12, 1, 1, 6, 16, 2, 1, 7, 20, 6, 1, 7, 25, 10, 1, 8, 30, 19, 1, 8, 36, 28, 1, 1, 9, 42, 44, 3, 1, 9, 49, 60, 9, 1, 10, 56, 85, 19, 1, 10, 64, 110, 38, 1, 11, 72, 146, 66, 1
Offset: 5

Views

Author

Christopher Hunt Gribble, Feb 23 2014

Keywords

Examples

			The first 9 rows of T(n,k) are:
.\ k    0      1      2
n
5       1      1
6       1      1
7       1      2
8       1      2
9       1      3
10      1      3      1
11      1      4      2
12      1      4      4
13      1      5      6

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Programs

  • Mathematica
    T[n_, k_] := (Binomial[n - 4k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 4k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 5, 20}, {k, 0, Quotient[n, 5]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
  • PARI
    T(n,k)={(binomial(n-4*k,k) + (k%2==0||n%2==1)*binomial((n-4*k-n%2)/2,k\2))/2}
for(n=5,20,for(k=0,(n\5), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms extended and xrefs updated by Christopher Hunt Gribble, Apr 26 2015
Terms a(27) and beyond from Andrew Howroyd, May 29 2017

A238189 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=n-n%2, read by rows.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 4, 7, 3, 1, 1, 4, 13, 10, 4, 1, 6, 23, 33, 22, 6, 1, 1, 6, 34, 68, 72, 30, 6, 1, 8, 49, 139, 204, 145, 54, 8, 1, 1, 8, 65, 230, 467, 476, 269, 70, 9, 1, 10, 85, 377, 961, 1348, 1080, 472, 111, 12, 1, 1, 10, 106, 552, 1767, 3188, 3454, 2156
Offset: 2

Views

Author

Christopher Hunt Gribble, Feb 19 2014

Keywords

Examples

			The first 10 rows of T(n,k) are:
  k  0     1     2     3     4     5     6     7     8     9    10
n
2    1     2     1
3    1     2     2
4    1     4     7     3     1
5    1     4    13    10     4
6    1     6    23    33    22     6     1
7    1     6    34    68    72    30     6
8    1     8    49   139   204   145    54     8     1
9    1     8    65   230   467   476   269    70     9
10   1    10    85   377   961  1348  1080   472   111    12     1
11   1    10   106   552  1767  3188  3454  2156   779   140    12

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Extensions

Terms corrected and extended by Christopher Hunt Gribble, Apr 25 2015

A238190 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 8, 1, 4, 12, 3, 1, 4, 18, 8, 1, 5, 24, 22, 1, 5, 32, 40, 6, 1, 6, 40, 73, 22, 1, 6, 50, 112, 66, 1, 7, 60, 172, 146, 10, 1, 7, 72, 240, 292, 48, 1, 8, 84, 335, 516, 174, 1, 8, 98, 440, 860, 448, 20
Offset: 3

Views

Author

Christopher Hunt Gribble, Feb 19 2014

Keywords

Examples

			The first 13 rows of T(n,k) are:
.\ k    0     1     2     3     4     5
n
3       1     1
4       1     1
5       1     2
6       1     2     2
7       1     3     4
8       1     3     8
9       1     4    12     3
10      1     4    18     8
11      1     5    24    22
12      1     5    32    40     6
13      1     6    40    73    22
14      1     6    50   112    66
15      1     7    60   172   146    10

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Programs

  • Mathematica
    T[n_, k_] := (2^k Binomial[n - 2k, k] + (Boole[EvenQ[k]] + Boole[OddQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k + 1, 2] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
  • PARI
    T(n,k)={(2^k*binomial(n-2*k,k) + ((k%2==0)+(n%2==1||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-2*k-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/3), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Link to C++ program and xrefs updated by Christopher Hunt Gribble, Apr 25 2015
Terms a(51) and beyond from Andrew Howroyd, May 29 2017

A238550 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=3*floor(n/2), read by rows.

Original entry on oeis.org

1, 3, 4, 1, 1, 3, 8, 3, 1, 6, 23, 33, 22, 6, 1, 1, 6, 40, 101, 125, 54, 10, 1, 9, 68, 262, 534, 532, 276, 74, 12, 1, 1, 9, 98, 509, 1551, 2505, 2196, 971, 219, 20, 1, 12, 139, 927, 3731, 8772, 12069, 9506, 4366, 1160, 179, 16, 1, 1, 12, 182, 1479, 7644, 24024
Offset: 2

Views

Author

Christopher Hunt Gribble, Feb 28 2014

Keywords

Examples

			The first 6 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
2     1     3     4     1
3     1     3     8     3
4     1     6    23    33    22     6     1
5     1     6    40   101   125    54    10
6     1     9    68   262   534   532   276    74    12     1
7     1     9    98   509  1551  2505  2196   971   219    20

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238551, A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

A238552 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 4, 1, 6, 4, 1, 6, 9, 1, 8, 18, 1, 8, 28, 1, 10, 42, 10, 1, 10, 57, 28, 1, 12, 76, 76, 1, 12, 96, 140, 1, 14, 120, 254, 25, 1, 14, 145, 392, 107, 1, 16, 174, 600, 321, 1, 16, 204, 840, 731, 1, 18, 238, 1170, 1462, 70, 1, 18, 273, 1540, 2610, 366
Offset: 4

Views

Author

Christopher Hunt Gribble, Feb 28 2014

Keywords

Examples

			The first 14 rows of T(n,k) are:
.\ k    0      1      2      3     4
n
4       1      2
5       1      2
6       1      4
7       1      4
8       1      6      4
9       1      6      9
10      1      8     18
11      1      8     28
12      1     10     42     10
13      1     10     57     28
14      1     12     76     76
15      1     12     96    140
16      1     14    120    254    25
17      1     14    145    392   107

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Programs

  • Mathematica
    T[n_, k_] := ((3^k + 1) Binomial[n - 3k, k] + Boole[EvenQ[k] || EvenQ[n]]*(3^Quotient[k + 1, 2] + 3^Quotient[k, 2]) * Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 4, 20}, {k, 0, Floor[n/4]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
  • PARI
    T(n,k)={((3^k+1)*binomial(n-3*k,k) + (k%2==0||n%2==0) * (3^((k+1)\2)+3^(k\2)) * binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=4,20,for(k=0,floor(n/4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 27 2015
Terms a(28) and beyond from Andrew Howroyd, May 29 2017

A238555 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=3*floor(n/2), read by rows.

Original entry on oeis.org

1, 3, 6, 2, 1, 3, 12, 8, 1, 6, 34, 68, 72, 30, 6, 1, 6, 58, 206, 403, 336, 106, 1, 9, 98, 509, 1551, 2505, 2196, 971, 219, 20, 1, 9, 140, 980, 4248, 10592, 15614, 12876, 5462, 900, 1, 12, 198, 1742, 9748, 33644, 73274, 98779, 80661, 37886, 9679, 1258, 72
Offset: 2

Views

Author

Christopher Hunt Gribble, Feb 28 2014

Keywords

Examples

			The first 6 rows of T(n,k) are:
./ k  0     1     2     3     4     5     6     7     8     9
n
2     1     3     6     2
3     1     3    12     8
4     1     6    34    68    72    30     6
5     1     6    58   206   403   336   106
6     1     9    98   509  1551  2505  2196   971   219    20
7     1     9   140   980  4248 10592 15614 12876  5462   900

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 27 2015

A238556 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=2*floor(n/3), read by rows.

Original entry on oeis.org

1, 3, 2, 1, 3, 4, 1, 6, 9, 1, 6, 21, 13, 4, 1, 9, 39, 53, 23, 1, 9, 64, 128, 87, 1, 12, 95, 283, 311, 91, 10, 1, 12, 133, 521, 891, 543, 106, 1, 15, 177, 917, 2118, 2030, 646, 1, 15, 228, 1444, 4424, 6000, 3295, 456, 25, 1, 18, 285, 2207, 8408, 15484, 13106, 4322, 473
Offset: 3

Views

Author

Christopher Hunt Gribble, Feb 28 2014

Keywords

Examples

			The first 5 rows of T(n,k) are:
./ k    0      1      2      3      4      5      6
n
3       1      3      2
4       1      3      4
5       1      6      9
6       1      6     21     13      4
7       1      9     39     53     23
8       1      9     64    128     87
9       1     12     95    283    311     91     10
10      1     12    133    521    891    543    106
11      1     15    177    917   2118   2030    646

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 27 2015
Terms a(48) and beyond by Andrew Howroyd, May 29 2017
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