A248011 -id:A248011 - cp's OEIS frontend

A248016 Sum over each antidiagonal of A248011.

0, 0, 3, 16, 67, 204, 546, 1268, 2714, 5348, 9965, 17580, 29781, 48520, 76660, 117624, 176196, 257976, 370503, 522456, 725175, 991540, 1337974, 1782924, 2349438, 3063164, 3955601, 5061524, 6423017, 8086224, 10106280, 12543280, 15468232, 18958128, 23103051, 28000224, 33762411, 40510812, 48384906, 57534052

Offset: 1

Christopher Hunt Gribble, Sep 29 2014

nonn

			a(1..9) are formed as follows:
.             Antidiagonals of A248011                n  a(n)
.                         0                            1     0
.                      0     0                         2     0
.                   1     1     1                      3     3
.                2     6     6     2                   4    16
.             6    14    27    14     6                5    67
.         10    32    60    60    32    10             6   204
.      19    55   129   140   129    55    19          7   546
.   28    94   218   294   294   218    94    28       8  1268
.44   140   363   506   608   506   363   140    44    9  2714

Christopher Hunt Gribble, Table of n, a(n) for n = 1..10000

Cf. A248011.

b := proc (n::integer, k::integer)::integer;
(4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;
end proc;
for j to 10000 do
    a := 0;
    for k from j by -1 to 1 do
       n := j-k+1;
       a := a+b(n, k)
    end do;
    printf("%d, ", a)
end do;

Empirically, a(n) = (2*n^7 + 14*n^6 + 14*n^5 + 70*n^4 - 77*n^3 - 399*n^2 + 61*n + 105 - 105*(-1)^n - 35*n^3*(-1)^n - 105*n^2*(-1)^n + 35*n*(-1)^n)/6720.

Empirical g.f.: -x^3*(x^2+1)*(x^4-6*x^2-4*x-3) / ((x-1)^8*(x+1)^4). - Colin Barker, Apr 06 2015

Terms corrected and extended by Christopher Hunt Gribble, Apr 02 2015

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

A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 6, 6, 4, 6, 10, 13, 10, 6, 9, 15, 22, 22, 15, 9, 12, 21, 34, 36, 34, 21, 12, 16, 28, 48, 56, 56, 48, 28, 16, 20, 36, 65, 78, 88, 78, 65, 36, 20, 25, 45, 84, 106, 123, 123, 106, 84, 45, 25, 30, 55, 106, 136, 168, 171, 168, 136, 106, 55, 30

Offset: 1

Christopher Hunt Gribble, Jun 25 2014

			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     0    1    2    4    6    9   12   16   20   25   30
.2     1    3    6   10   15   21   28   36   45   55   66
.3     2    6   13   22   34   48   65   84  106  130  157
.4     4   10   22   36   56   78  106  136  172  210  254
.5     6   15   34   56   88  123  168  216  274  335  406
.6     9   21   48   78  123  171  234  300  381  465  564
.7    12   28   65  106  168  234  321  412  524  640  777
.8    16   36   84  136  216  300  412  528  672  820  996
.9    20   45  106  172  274  381  524  672  856 1045 1270
10    25   55  130  210  335  465  640  820 1045 1275 1550
11    30   66  157  254  406  564  777  996 1270 1550 1885

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244307, A248011, A248016, A248059, A248060, A248017, A248027.

Empirically,
T(n,k) = (4*k^2*n^2 + 2*k^2 + 8*k*n + 2*n^2 - 4*k - 4*n - 1 - (2*k^2 - 4*k - 1)*(-1)^n - (2*n^2 - 4*n - 1)*(-1)^k - (-1)^k*(-1)^n)/32.
T(1,k) = A002620(k) = floor(k^2/4).
T(2,k) = A000217(k) = k*(k+1)/2.
       = T(1,k) + T(1,k+1) = floor(k^2/4) + floor((k+1)^2/4).
T(3,k) = 2*A000217(k) + A024206(k-2)
       = k*(k+1) + floor((k-1)^2/4) - 1.

Terms corrected and extended by Christopher Hunt Gribble, Apr 02 2015

A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 14, 39, 14, 1, 3, 66, 208, 208, 66, 3, 12, 198, 794, 1092, 794, 198, 12, 28, 508, 2196, 3912, 3912, 2196, 508, 28, 66, 1092, 5231, 10626, 13462, 10626, 5231, 1092, 66, 126, 2156, 10808, 24648, 35787, 35787, 24648, 10808, 2156, 126

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=8 and 1<=k<=8 is:
.  k   1      2      3      4      5      6      7       8 ...
n
1      0      0      0      0      1      3     12      28
2      0      0      2     14     66    198    508    1092
3      0      2     39    208    794   2196   5231   10808
4      0     14    208   1092   3912  10626  24648   50344
5      1     66    794   3912  13462  35787  81648  164980
6      3    198   2196  10626  35787  94248 212988  428076
7     12    508   5231  24648  81648 212988 477903  955856
8     28   1092  10808  50344 164980 428076 955856 1906128

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
   + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
   - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
   - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
   + 48*k + 48*n + 45
   + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
      + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
      - 45)*(-1)^k
   + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
      - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n
      - 48*k - 45)*(-1)^n
   + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4 - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3 - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2 + 48*k + 48*n + 45
+ (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3 + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n - 45)*(-1)^k
+ (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k - 45)*(-1)^n
+ (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;
T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.

Terms corrected and extended by Christopher Hunt Gribble, Apr 16 2015

A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 6, 6, 1, 3, 22, 39, 22, 3, 9, 60, 139, 139, 60, 9, 19, 135, 371, 476, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1      2      3      4      5      6      7      8       9 ...
n
1       0      0      0      1      3      9     19     38      66
2       0      1      6     22     60    135    266    476     792
3       0      6     39    139    371    813   1574   2770    4554
4       1     22    139    476   1253   2706   5199   9080   14857
5       3     60    371   1253   3254   6969  13294  23102   37637
6       9    135    813   2706   6969  14841  28197  48852   79401
7      19    266   1574   5199  13294  28197  53381  92266  149645
8      38    476   2770   9080  23102  48852  92266 159216  257878
9      66    792   4554  14857  37637  79401 149645 257878  417156

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384;
T(1,k) = sum(A005993(i-4),i=1,k)
       = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

cp's OEIS Frontend

A248016 Sum over each antidiagonal of A248011.

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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.

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A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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