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 22 results. Next

A228168 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (9,n)-rectangular grid with k '1's and (9n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 5, 20, 44, 66, 66, 44, 20, 5, 1, 1, 5, 45, 208, 792, 2156, 4704, 7984, 11034, 12190, 11034, 7984, 4704, 2156, 792, 208, 45, 5, 1, 1, 10, 106, 785, 4554, 20556, 74828, 223498, 557599, 1175574, 2114550, 3266415, 4354188, 5023464, 5023464, 4354188
Offset: 0

Views

Author

Yosu Yurramendi, María Merino, Aug 14 2013

Keywords

Comments

The length of row n is 9*n+1.
Sum of rows (see example) gives A225832.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, triangle A228165 to A225830, triangle A228166 to A225831, and triangle A228167 to A225832.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 9 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 27 2015

Examples

			Irregular triangle:
1
1  5  20  44   66    66    44     20      5       1
1  5  45 208  792  2156  4704   7984  11034   12190   11034    7984  ...
1 10 106 785 4554 20556 74828 223498 557599 1175574 2114550 3266415 4354188 5023464 5023464 4354188 3266415 2114550 1175574 ...

Links

Crossrefs

Cf. A225826-A225832, A005418, A034851, A226048, A226290, A225812, A228022, A228165, A228166, A228167.

Extensions

Definition corrected by María Merino, May 22 2017

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

A231568 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 5 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, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 4, 1, 4, 8, 1, 4, 12, 1, 5, 18, 3, 1, 5, 24, 8, 1, 6, 32, 22, 1, 6, 40, 40, 1, 7, 50, 73, 6, 1, 7, 60, 112, 22, 1, 8, 72, 172, 66, 1, 8, 84, 240, 146, 1, 9, 98, 335, 292, 10, 1, 9, 112, 440, 516, 48, 1, 10, 128, 578, 860, 174
Offset: 4

Views

Author

Christopher Hunt Gribble, Feb 23 2014

Keywords

Examples

			The first 14 rows of T(n,k) are:
.\  k    0      1      2      3     4
n
4        1      1
5        1      1
6        1      2
7        1      2
8        1      3      2
9        1      3      4
10       1      4      8
11       1      4     12
12       1      5     18      3
13       1      5     24      8
14       1      6     32     22
15       1      6     40     40
16       1      7     50     73     6
17       1      7     60    112    22

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, 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 - 3k, k] + (Boole[EvenQ[k]] + Boole[EvenQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k+1, 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)={(2^k*binomial(n-3*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/4), 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(32) 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
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