A228572 -id:A228572 - cp's OEIS frontend

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.

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

A192928 The Gi1 and Gi2 sums of Losanitsch's triangle A034851.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11, 16, 20, 29, 37, 53, 69, 98, 130, 183, 245, 343, 463, 646, 877, 1220, 1664, 2310, 3161, 4381, 6009, 8319, 11430, 15811, 21751, 30070, 41405, 57216, 78836, 108906, 150130, 207346

Offset: 0

Johannes W. Meijer, Jul 14 2011

The Gi1 and Gi2 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal this sequence.
From Johannes W. Meijer, Aug 26 2013: (Start)
The a(n) are also the Ca1 and Ca2 sums of McGarvey’s triangle A102541.
Furthermore they are the Kn11 and Kn12 sums of triangle A228570.
And finally the terms of this sequence are the row sums of triangle A228572. (End)

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 25.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,0,-1,0,1,-1,0,0,-1).

Cf. A003269, A034851, A102541, A180662, A228570, A228572.

A192928 := proc(n): (A003269(n+1)+x(n)+x(n-1)+x(n-4))/2 end: A003269 := proc(n): sum(binomial(n-1-3*j, j), j=0..(n-1)/3) end: x:=proc(n): if type(n,even) then A003269(n/2+1) else 0 fi: end: seq(A192928(n),n=0..42);

LinearRecurrence[{1, 1, -1, 1, 0, -1, 0, 1, -1, 0, 0, -1}, {1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11}, 43] (* Jean-François Alcover, Nov 16 2017 *)

G.f.: (-1/2)*(1/(x^4+x-1) + (1+x+x^4)/(x^8+x^2-1))= -(1+x)*(x^7-x^6+x^5+x-1) / ( (x^4+x-1)*(x^8+x^2-1) ).
a(n) = (A003269(n+1)+x(n)+x(n-1)+x(n-4))/2 with x(2*n) = A003269(n+1) and x(2*n+1) = 0.
From Johannes W. Meijer, Aug 26 2013: (Start)
a(n) = sum(A228572(n, k), k=0..n)
a(n) = sum(A228570(n-k, k), k=0..floor(n/2))
a(n) = sum(A102541(n-2*k, k), k=0..floor(n/3))
a(n) = sum(A034851(n-3*k, k), k=0..floor(n/4)) (End)

A180184 Irregular triangle read by rows: T(n,k) is the number of compositions of n with k parts, all >= 4, for n >= 4 and 1 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 1, 6, 3, 1, 7, 6, 1, 8, 10, 1, 9, 15, 1, 1, 10, 21, 4, 1, 11, 28, 10, 1, 12, 36, 20, 1, 13, 45, 35, 1, 1, 14, 55, 56, 5, 1, 15, 66, 84, 15, 1, 16, 78, 120, 35, 1, 17, 91, 165, 70, 1, 1, 18, 105, 220, 126, 6, 1, 19, 120, 286, 210, 21, 1, 20

Offset: 4

Emeric Deutsch, Aug 15 2010

Sum of the entries in row n is A003269(n-3).
Row n contains floor(n/4) entries.
From Petros Hadjicostas, Apr 15 2020: (Start)
From equations (3) and (23) in Mathar (2016), we have Sum_{m,k >= 0} T_{4 x m}(4,k)*z^m*t^k = 1/(1 - z - z^4*t), where T_{4xm}(4,k) counts the ways of tiling the 4 x m rectangle with k non-overlapping squares of shape 4 x 4 and with 4*m - 16*k unit squares (see Definition 1 in his paper for more details).
By expanding 1/(1 - (z + z^4*t)) as an infinite geometric series, using the binomial theorem, and changing indices of summation, we may show that T_{4xm}(4,k) = binomial(m - 3*k, k) = T(m+4, k+1) for m >= 0 and 0 <= k <= floor(m/4) (where the current array T(n,k) should be distinguished from Mathar's T_{4 x m}(4,k)).
Indeed, we have a bijection between the above tilings of the 4 x m rectangle with k non-overlapping squares of shape 4 x 4 and 4*m - 16*k unit squares and the compositions of m+4 with k+1 parts, all >= 4. To construct the bijection, let us agree that an a x b rectangle has height a and base b.
Given such a composition, a_1 + a_2 + ... + a_{k+1} = m + 4 (with a_i >= 4), paste together a 4 x 4 square followed by a_1 - 4 columns of 4 unit squares, another 4 x 4 square followed by a_2 - 4 columns of 4 units squares, and so on, and finally a 4 x 4 square followed by a_{k+1} - 4 columns of 4 unit squares. Remove the first 4 x 4 square, and we get a tiling of the 4 x m rectangle with k 4 x 4 squares and 4*Sum_{i=1..(k+1)} (a_i - 4) = 4*(m+4) - 16*(k+1) = 4*m - 16*k unit squares.
The above process can be reversed to complete the bijection, but we omit the details. (End)
T(n+7,k+1) is the number of k-subsets of {1..n} with values at least 4 apart. For example, T(17,4) = 4 corresponds to the subsets {1,5,9},{1,5,10},{1,6,10},{2,6,10} of {1..10} (A102547 gives the number of k-subsets of {1..n} with values at least 3 apart and A011973 with values at least 2 apart). - Enrique Navarrete, Jan 29 2022

			Triangle T(n,k) (with n >= 4 and 1 <= k <= floor(n/4)) starts as follows:
  1;
  1;
  1;
  1;
  1,  1;
  1,  2;
  1,  3;
  1,  4;
  1,  5,  1;
  1,  6,  3;
  1,  7,  6;
  1,  8, 10;
  1,  9, 15,  1;
  1, 10, 21,  4;
  1, 11, 28, 10;
  1, 12, 36, 20;
  ...
T(14,3) = 6 because we have the following compositions (ordered partitions) of 14 with 3 parts, all >= 4: [5,5,4], [4,6,4], [5,4,5], [6,4,4], [4,5,5], [4,4,6].

R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.3.

Cf. A003269, A102547, A228572.

for n from 4 to 27 do seq(binomial(n-3*k-1, k-1), k = 1 .. floor((1/4)*n)) end do;
T := (n,k) -> binomial(n-3*k-1, k-1): seq(seq(T(n,k), k=1..floor(n/4)), n=4..26); # Johannes W. Meijer, Aug 26 2013

Flatten[Table[Binomial[n-3k-1,k-1],{n,4,30},{k,Floor[n/4]}]] (* Harvey P. Dale, Feb 05 2013 *)

T(n, k) = binomial(n-3*k-1, k-1).
T(n, k) = A228572(2*n-4, 2*k-1) + A228572(2*n-7, 2*k-2) - A228572(2*n-3, 2*k-1) for n >= 4 and 1 <= k <= floor(n/4). - Johannes W. Meijer, Aug 26 2013 [Range of k adjusted by Petros Hadjicostas, Apr 15 2020 to start at 1 rather than 0]
G.f.: Sum_{n,k} T(n,k)*z^n*t^k = z^4*t/(1-z-t*z^4). - R. J. Mathar, Aug 24 2016 [Adjusted by Petros Hadjicostas, Apr 14 2020 to agree with the offset]

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

Christopher Hunt Gribble, Apr 27 2015

			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

Andrew Howroyd, Table of n, a(n) for n = 4..860
Christopher Hunt Gribble, C++ program

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

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

Terms a(24) and beyond by Andrew Howroyd, May 29 2017

cp's OEIS Frontend

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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A192928 The Gi1 and Gi2 sums of Losanitsch's triangle A034851.

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A180184 Irregular triangle read by rows: T(n,k) is the number of compositions of n with k parts, all >= 4, for n >= 4 and 1 <= k <= floor(n/4).

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Maple

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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.

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