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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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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

Views

Author

Christopher Hunt Gribble, Sep 30 2014

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
    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);

Formula

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.

Extensions

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

Views

Author

Christopher Hunt Gribble, Sep 30 2014

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
    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);

Formula

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.

Extensions

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

A226292 (10*n^2+4*n+(1-(-1)^n))/8.

Original entry on oeis.org

2, 6, 13, 22, 34, 48, 65, 84, 106, 130, 157, 186, 218, 252, 289, 328, 370, 414, 461, 510, 562, 616, 673, 732, 794, 858, 925, 994, 1066, 1140, 1217, 1296, 1378, 1462, 1549, 1638, 1730, 1824, 1921, 2020, 2122, 2226, 2333, 2442, 2554, 2668, 2785, 2904, 3026, 3150
Offset: 1

Views

Author

Yosu Yurramendi, Jun 02 2013

Keywords

Comments

The number of binary pattern classes in the (3,n)-rectangular grid with 2 '1's and (n-2) '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, n<10.
Column k=2 of A226290.
For n even, a(n) is A202803; for n odd, a(n) is A190816.
Number of lattice points (x,y) in the region bounded by y < 3x, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 31 2014

Links

Crossrefs

Cf. A190816, A202803, A226290.

Programs

  • Magma
    [(10*n^2+4*n+(1-(-1)^n))/8: n in [1..50]]; // Vincenzo Librandi, Sep 04 2013
  • Maple
    A226292:=n->(10*n^2+4*n+(1-(-1)^n))/8: seq(A226292(n), n=1..50); # Wesley Ivan Hurt, Oct 31 2014
  • Mathematica
    CoefficientList[Series[(2 + 2 x + x^2) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 04 2013 *)
LinearRecurrence[{2,0,-2,1},{2,6,13,22},60] (* Harvey P. Dale, Feb 01 2019 *)

Formula

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4, a(1)=2, a(2)=6, a(3)=13, a(4)=22.
a(n) = 2*a(n-2)-a(n-4)+10 for n>4, a(1)=2, a(2)=6, a(3)=13, a(4)=22.
a(n) = a(n-1)+a(n-2)-a(n-3)+5 for n>3, a(1)=2, a(2)=6, a(3)=13.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+(-1)^n for n>3, a(1)=2, a(2)=6, a(3)=13.
a(n) = 2*a(n-1)-a(n-2)+2+(1-(-1)^n)/2 for n>2, a(1)=2, a(2)=6.
G.f.: x*(2+2*x+x^2)/((1+x)*(1-x)^3). - Bruno Berselli, Jun 03 2013

Extensions

More terms from Vincenzo Librandi, Sep 04 2013

A257523 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 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, 6, 1, 6, 14, 1, 8, 28, 1, 8, 44, 1, 10, 66, 20, 1, 10, 90, 64, 1, 12, 120, 168, 1, 12, 152, 320, 1, 14, 190, 572, 72, 1, 14, 230, 896, 328, 1, 16, 276, 1360, 984, 1, 16, 324, 1920, 2264, 1, 18, 378, 2660, 4528, 272
Offset: 4

Views

Author

Christopher Hunt Gribble, Apr 27 2015

Keywords

Examples

			The first 9 rows of T(n,k) are:
.\ k    0      1      2     3
n
4       1      2
5       1      2
6       1      4
7       1      4
8       1      6      6
9       1      6     14
10      1      8     28
11      1      8     44
12      1     10     66    20
13      1     10     90    64
14      1     12    120   168
15      1     12    152   320

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, 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)={(4^k*binomial(n-3*k,k) + ((n%2==0||k%2==0)+(k%2==0)+(k==0)) * 4^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=4,15,for(k=0,(n\4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms a(24) and beyond by Andrew Howroyd, May 29 2017
Previous Showing 31-34 of 34 results.

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