A226048 -id:A226048 - cp's OEIS frontend

A238559 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 8 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=2*floor(n/4), read by rows.

1, 3, 1, 1, 3, 2, 1, 6, 4, 1, 6, 6, 1, 9, 17, 5, 1, 1, 9, 32, 18, 4, 1, 12, 56, 46, 13, 1, 12, 84, 90, 31, 1, 15, 121, 193, 98, 13, 1, 1, 15, 162, 360, 275, 66, 6, 1, 18, 212, 664, 672, 250, 31, 1, 18, 266, 1080, 1408, 672, 110, 1, 21, 329, 1711, 2797, 1772, 432, 23, 1

Offset: 4

Christopher Hunt Gribble, Feb 28 2014

			The first 11 rows of T(n,k) are:
.\  k   0      1      2      3      4     5     6
n
4       1      3      1
5       1      3      2
6       1      6      4
7       1      6      6
8       1      9     17      5      1
9       1      9     32     18      4
10      1     12     56     46     13
11      1     12     84     90     31
12      1     15    121    193     98    13     1
13      1     15    162    360    275    66     6
14      1     18    212    664    672   250    31

Andrew Howroyd, Table of n, a(n) for n = 4..992
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557, A238558, A228168, A238581-A238583, A228169, A238586, A238592.

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

Terms a(31) and beyond from Andrew Howroyd, May 29 2017

A238581 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 9 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=4*floor(n/2), read by rows.

Original entry on oeis.org

1, 4, 12, 10, 3, 1, 4, 24, 40, 22, 1, 8, 65, 230, 467, 476, 269, 70, 9, 1, 8, 109, 641, 2281, 4424, 4718, 2409, 473, 1, 12, 182, 1479, 7644, 24024, 47022, 56226, 41000, 17834, 4545, 625, 39, 1, 12, 258, 2762, 19347, 86536, 255552, 495547, 625705, 499314, 239254, 61732, 6533

Offset: 2

Christopher Hunt Gribble, Mar 01 2014

			The first 4 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8
n
2     1     4    12    10     3
3     1     4    24    40    22
4     1     8    65   230   467   476   269    70     9
5     1     8   109   641  2281  4424  4718  2409   473

Andrew Howroyd, Table of n, a(n) for n = 2..991
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238582, A238583, A228169, A238586, A238592.

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

Terms a(43) and beyond from Andrew Howroyd, May 29 2017

A238583 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 9 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=2*floor(n/4), read by rows.

Original entry on oeis.org

1, 3, 2, 1, 3, 4, 1, 6, 9, 1, 6, 14, 1, 9, 32, 18, 4, 1, 9, 55, 65, 23, 1, 12, 91, 164, 87, 1, 12, 132, 320, 229, 1, 15, 186, 608, 648, 134, 10, 1, 15, 245, 1043, 1633, 770, 106, 1, 18, 317, 1736, 3659, 2800, 646, 1, 18, 394, 2666, 7247, 7572, 2510

Offset: 4

Christopher Hunt Gribble, Mar 01 2014

			The first 8 rows of T(n,k) are:
.\ k    0      1      2      3      4
n
4       1      3      2
5       1      3      4
6       1      6      9
7       1      6     14
8       1      9     32     18      4
9       1      9     55     65     23
10      1     12     91    164     87
11      1     12    132    320    229

Andrew Howroyd, Table of n, a(n) for n = 4..992
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581, A238582, A228169, A238586, A238592

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

Terms a(26) and beyond from Andrew Howroyd, May 29 2017

A238586 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 10 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=5*floor(n/2), read by rows.

Original entry on oeis.org

1, 5, 16, 19, 9, 1, 1, 5, 32, 73, 66, 10, 1, 10, 85, 377, 961, 1348, 1080, 472, 111, 12, 1, 1, 10, 142, 1011, 4429, 11370, 17252, 14478, 6094, 1020, 70, 1, 15, 236, 2280, 14203, 56571, 146212, 244063, 261847, 179063, 77974, 21422, 3637, 368, 24, 1

Offset: 2

Christopher Hunt Gribble, Mar 01 2014

			The first 4 rows of T(n,k) are:
.\k  0     1     2     3     4     5     6     7     8     9    10
n
2    1     5    16    19     9     1
3    1     5    32    73    66    10
4    1    10    85   377   961  1348  1080   472   111    12     1
5    1    10   142  1011  4429 11370 17252 14478  6094  1020    70

Andrew Howroyd, Table of n, a(n) for n = 2..937
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238592.

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

Terms a(36) and beyond from Andrew Howroyd, May 29 2017

A238592 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 10 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=3*floor(n/3), read by rows.

Original entry on oeis.org

1, 4, 9, 2, 1, 4, 18, 8, 1, 8, 42, 28, 1, 8, 77, 165, 151, 44, 6, 1, 12, 133, 521, 891, 543, 106, 1, 12, 200, 1160, 3022, 2756, 824, 1, 16, 288, 2260, 8443, 13336, 9364, 2819, 387, 20, 1, 16, 387, 3867, 19833, 48418, 58731, 34797, 9462, 900

Offset: 3

Christopher Hunt Gribble, Mar 01 2014

			The first 6 rows of T(n,k) are:
.\ k    0      1      2      3      4      5      6
n
3       1      4      9      2
4       1      4     18      8
5       1      8     42     28
6       1      8     77    165    151     44      6
7       1     12    133    521    891    543    106
8       1     12    200   1160   3022   2756    824

Andrew Howroyd, Table of n, a(n) for n = 3..989
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586.

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

Terms a(36) and beyond from Andrew Howroyd, May 29 2017

A238558 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 8 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, 4, 1, 3, 8, 1, 6, 18, 1, 6, 36, 32, 13, 1, 9, 64, 128, 87, 1, 9, 100, 308, 332, 1, 12, 146, 647, 1118, 451, 68, 1, 12, 200, 1160, 3022, 2756, 824, 1, 15, 264, 1958, 6882, 10076, 5009, 1, 15, 336, 3020, 13798, 28774, 24237, 4774, 346

Offset: 3

Christopher Hunt Gribble, Feb 28 2014

			The first 8 rows of T(n,k) are:
.\ k    0      1      2      3      4      5      6
n
3       1      3      4
4       1      3      8
5       1      6     18
6       1      6     36     32     13
7       1      9     64    128     87
8       1      9    100    308    332
9       1     12    146    647   1118    451     68
10      1     12    200   1160   3022   2756    824

Andrew Howroyd, Table of n, a(n) for n = 3..971
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557, A238559, A228168, A238581-A238583, A228169, A238586, A238592.

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

Terms a(41) and beyond from Andrew Howroyd, May 29 2017

A238582 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 9 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, 4, 6, 1, 1, 4, 12, 3, 1, 8, 28, 10, 1, 8, 54, 82, 49, 8, 1, 1, 12, 95, 283, 311, 91, 10, 1, 12, 146, 647, 1118, 451, 68, 1, 16, 212, 1300, 3380, 3076, 1200, 209, 20, 1, 1, 16, 288, 2260, 8443, 13336, 9364, 2819, 387, 20

Offset: 3

Christopher Hunt Gribble, Mar 01 2014

			The first 9 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
3     1     4     6     1
4     1     4    12     3
5     1     8    28    10
6     1     8    54    82    49     8     1
7     1    12    95   283   311    91    10
8     1    12   146   647  1118   451    68
9     1    16   212  1300  3380  3076  1200   209    20     1
10    1    16   288  2260  8443 13336  9364  2819   387    20
11    1    20   379  3709 18203 42412 44599 19051  3682   282

Andrew Howroyd, Table of n, a(n) for n = 3..989
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581, A238583, A228169, A238586, A238592.

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

Terms a(46) and beyond from Andrew Howroyd, May 29 2017

A238551 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 6 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, 2, 1, 1, 2, 2, 1, 4, 4, 1, 4, 11, 3, 1, 1, 6, 21, 13, 4, 1, 6, 36, 32, 13, 1, 8, 54, 82, 49, 8, 1, 1, 8, 77, 165, 151, 44, 6, 1, 10, 103, 319, 382, 173, 31, 1, 10, 134, 530, 867, 559, 164, 12, 1, 1, 12, 168, 852, 1789, 1632, 705, 119, 9

Offset: 3

Christopher Hunt Gribble, Feb 28 2014

			The first 12 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8
n
3     1     2     1
4     1     2     2
5     1     4     4
6     1     4    11     3     1
7     1     6    21    13     4
8     1     6    36    32    13
9     1     8    54    82    49     8     1
10    1     8    77   165   151    44     6
11    1    10   103   319   382   173    31
12    1    10   134   530   867   559   164    12     1
13    1    12   168   852  1789  1632   705   119     9
14    1    12   207  1255  3409  4074  2406   618    66

Andrew Howroyd, Table of n, a(n) for n = 3..971
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

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

Terms a(57) and beyond from Andrew Howroyd, May 29 2017

A071239 a(n) = n(n+1)(n^2 + 2)/6.

Original entry on oeis.org

0, 1, 6, 22, 60, 135, 266, 476, 792, 1245, 1870, 2706, 3796, 5187, 6930, 9080, 11696, 14841, 18582, 22990, 28140, 34111, 40986, 48852, 57800, 67925, 79326, 92106, 106372, 122235, 139810, 159216, 180576, 204017, 229670, 257670, 288156, 321271, 357162, 395980

Offset: 0

N. J. A. Sloane, Jun 12 2002

Number of binary pattern classes with 4 black beads in the (2,n)-rectangular grid; two patterns are in the same class if one of them can be obtained by reflection or rotation of the other one. - Yosu Yurramendi, Sep 12 2008

This sequence is the case k=n+3 of b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6, which is the n-th k-gonal pyramidal number. Therefore, apart from 0, this sequence is the 3rd diagonal of the array in A080851. - Luciano Ancora, Apr 10 2015

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A080851, A226048.

[n*(n+1)*(n^2+2)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Table[(n(n+1)(n^2+2))/6,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,6,22,60},40] (* Harvey P. Dale, May 01 2013 *)

a(n)=n*(n+1)*(n^2+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

a <- vector()
    for(n in 1:40) a[n] <- (1/4)*(choose(2*n,4) + 3*choose(n,2))
    a
# Yosu Yurramendi and María Merino, Aug 21 2013

def A071239(n): return binomial(n+1,2)*(n^2+2)//3
[A071239(n) for n in range(41)] # G. C. Greubel, Aug 06 2024

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n>4. - Harvey P. Dale, May 01 2013
a(n) = (binomial(2*n+2,4) + 3*binomial(n+1,2))/4 = (A053134(n-1) + 3*A000217(n))/4 . - Yosu Yurramendi and María Merino, Aug 21 2013
G.f.: x*(1+x+2*x^2) / (1-x)^5 . - R. J. Mathar, Aug 21 2013
E.g.f.: (1/6)*x*(6 + 12*x + 7*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 06 2024

A243866 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing one 1 X 1 tile in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 3, 3, 4, 4, 3, 3, 4, 3, 6, 4, 6, 3, 4, 4, 4, 6, 6, 6, 6, 4, 4, 5, 4, 8, 6, 9, 6, 8, 4, 5, 5, 5, 8, 8, 9, 9, 8, 8, 5, 5, 6, 5, 10, 8, 12, 9, 12, 8, 10, 5, 6, 6, 6, 10, 10, 12, 12, 12, 12, 10, 10, 6, 6, 7, 6, 12, 10, 15

Offset: 1

Christopher Hunt Gribble, Jun 19 2014

It appears that the number of equivalence classes of ways of placing one m X m tile in an n X k rectangle under all symmetry operations of the rectangle is T(n-m+1,k-m+1) for m >= 2, n >= m and k >= m, and zero otherwise. - Christopher Hunt Gribble, Oct 17 2014
The sum over each antidiagonal of A243866
= Sum_{j=1..n}(2*j + 1 - (-1)^j)*(2*(n - j + 1) + 1 - (-1)^(n - j + 1))/16
= (n + 2)*(2*n^2 + 8*n + 3 - 3*(-1)^n)/48
- see A006918. - Christopher Hunt Gribble, Apr 01 2015

			T(n,k) for 1<=n<=11 and 1<=k<=11 is:
    k    1    2    3    4    5    6    7    8    9   10   11 ...
.n
.1       1    1    2    2    3    3    4    4    5    5    6
.2       1    1    2    2    3    3    4    4    5    5    6
.3       2    2    4    4    6    6    8    8   10   10   12
.4       2    2    4    4    6    6    8    8   10   10   12
.5       3    3    6    6    9    9   12   12   15   15   18
.6       3    3    6    6    9    9   12   12   15   15   18
.7       4    4    8    8   12   12   16   16   20   20   24
.8       4    4    8    8   12   12   16   16   20   20   24
.9       5    5   10   10   15   15   20   20   25   25   30
10       5    5   10   10   15   15   20   20   25   25   30
11       6    6   12   12   18   18   24   24   30   30   36
...

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248017, A248027.

b := proc (n,k);
floor((1/2)*n+1/2)*floor((1/2)*k+1/2)
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = floor((n+1)/2)*floor((k+1)/2)
       = (2*n+1-(-1)^n)*(2*k+1-(-1)^k)/4;
T(n,1) = A034851(n,1);
T(n,2) = A226048(n,1);
T(n,3) = A226290(n,1);
T(n,4) = A225812(n,1);
T(n,5) = A228022(n,1);
T(n,6) = A228165(n,1);
T(n,7) = A228166(n,1). - Christopher Hunt Gribble, May 01 2015

Terms corrected by Christopher Hunt Gribble, Mar 27 2015

cp's OEIS Frontend

A238559 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 8 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=2*floor(n/4), read by rows.

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A238581 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 9 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=4*floor(n/2), read by rows.

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A238583 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 9 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=2*floor(n/4), read by rows.

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A238586 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 10 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=5*floor(n/2), read by rows.

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A238592 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 10 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=3*floor(n/3), read by rows.

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A238558 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 8 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=2*floor(n/3), read by rows.

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A238582 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 9 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=2*floor(n/3), read by rows.

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A238551 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=2*floor(n/3), read by rows.

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A071239 a(n) = n*(n+1)*(n^2 + 2)/6.

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Magma

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Formula

A071239 a(n) = n(n+1)(n^2 + 2)/6.