A090642 -id:A090642 - cp's OEIS frontend

A098487 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square array such that all positions are mutually isolated. Two positions (s,t),(u,v) are considered as isolated from each other if min(abs(s-u),abs(t-v))>1.

1, 4, 0, 9, 16, 8, 16, 78, 140, 79, 25, 228, 964, 1987, 1974, 36, 520, 3920, 16834, 42368, 62266, 49, 1020, 11860, 85275, 397014, 1220298, 2484382, 64, 1806, 29708, 317471, 2326320, 12033330, 44601420, 119138166, 81, 2968, 65240, 962089, 10087628, 77784658, 450193818, 1979541332, 6655170642

Offset: 1

Hugo Pfoertner, Sep 15 2004

For more information, links, programs see A098485.

			T(3,3) = a(6) = 8 because there are the following 8 ways to pick 3 positions isolated from each other from a 3 X 3 square array:
X0X...X0X...X0X...X00...X00...0X0...00X...00X
000...000...000...00X...000...000...X00...000
X00...0X0...00X...X00...X0X...X0X...00X...X0X
Triangle begins:
:  1;
:  4,    0;
:  9,   16,     8;
: 16,   78,   140,     79;
: 25,  228,   964,   1987,    1974;
: 36,  520,  3920,  16834,   42368,    62266;
: 49, 1020, 11860,  85275,  397014,  1220298,  2484382;
: 64, 1806, 29708, 317471, 2326320, 12033330, 44601420, 119138166;

Alois P. Heinz, Rows n = 1..21, flattened

A098485 gives selections where all marks are connected, A090642 gives total number of possible selections.
Main diagonal gives A201513.
Cf. A291716, A291717, A291718, A292152, A292153, A292154, A292155, A292156.

Fortran
```
! See link in A098485.
```

T(8,8) corrected by Alois P. Heinz, May 11 2017

A348129 Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 8, 0, 1, 16, 44, 24, 2, 1, 25, 140, 204, 82, 10, 1, 36, 340, 1024, 982, 248, 4, 1, 49, 700, 3628, 7002, 4618, 832, 40, 1, 64, 1288, 10320, 34568, 46736, 22708, 3192, 92, 1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352, 1, 100, 3480, 54400, 412596, 1535440, 2716096, 2119176, 636524, 56832, 724

Offset: 0

Alois P. Heinz, Oct 01 2021

			T(3,2) = 8:
  .-----. .-----. .-----. .-----. .-----. .-----. .-----. .-----.
  |Q . .| |Q . .| |. . Q| |. . Q| |. . .| |. Q .| |. Q .| |. . .|
  |. . Q| |. . .| |. . .| |Q . .| |Q . .| |. . .| |. . .| |. . Q|
  |. . .| |. Q .| |. Q .| |. . .| |. . Q| |. . Q| |Q . .| |Q . .|
  `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,    0;
  1,  9,    8,     0;
  1, 16,   44,    24,      2;
  1, 25,  140,   204,     82,     10;
  1, 36,  340,  1024,    982,    248,      4;
  1, 49,  700,  3628,   7002,   4618,    832,     40;
  1, 64, 1288, 10320,  34568,  46736,  22708,   3192,    92;
  1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352;
  ...

Columns k=0-8 give: A000012, A000290, A036464, A047659, A061994, A108792, A176186, A178721, A252593.
Main diagonal gives A000170.
Row sums give A287227.
T(2n,n) gives A348130.
Cf. A090642, A144084, A178717.

A098485 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square array such that their adjacency graph consists of a single component. Two positions (s,t), (u,v) are considered as adjacent if max(abs(s-u), abs(t-v)) <= 1.

Original entry on oeis.org

1, 4, 6, 9, 20, 48, 16, 42, 132, 419, 25, 72, 256, 973, 3682, 36, 110, 420, 1747, 7484, 31992, 49, 156, 624, 2741, 12562, 58620, 273556, 64, 210, 868, 3955, 18916, 92912, 462104, 2927505, 81, 272, 1152, 5389, 26546, 134868, 697836, 3644935, 19082018

Offset: 1

Hugo Pfoertner, Sep 14 2004

Number of ways to mark the numbers on a square board on a lottery play slip such that one connected graphic pattern is formed. For the lottery "mark 6 numbers of 49 on a 7 X 7 grid of numbers" that is played in many countries, there are T(7,6)=58620 (out of binomial(49,6)=13983816) different combinations of 6 numbers whose graphic pattern on the board forms one connected component.

			a(5)=T(3,2)=20 because there are 20 ways to mark two positions in a 3 X 3 square grid such that the two picked positions are either row-wise, column-wise or diagonally adjacent:
XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000
000...X00...0X0...000...X00...0X0...00X...0X0...00X...XX0
000...000...000...000...000...000...000...000...000...000
.........................................................
000...000...000...000...000...000...000...000...000...000
000...X00...0X0...000...X00...0X0...00X...0X0...00X...0XX
XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000

John Burkardt, GRAFPACK Graph Computations.
Hugo Pfoertner, Counts of connected components in selected numbers on square lotto boards.
Hugo Pfoertner, Program to analyze the adjacency graph of selections on lotto boards.

Cf. A090642, A098487 (selections where all marks are isolated from each other), A291716, A291717, A291718, A292152, A292153, A292154, A292155, A292156.

Fortran
```
c See link.
```

A291717 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry.

Original entry on oeis.org

1, 4, 6, 9, 36, 8, 16, 120, 24, 168, 25, 300, 72, 714, 178, 36, 630, 144, 2273, 464, 6576, 49, 1176, 288, 5932, 1476, 24288, 6404, 64, 2016, 480, 13536, 3040, 74560, 15680, 341320, 81, 3240, 800, 27860, 6940, 197600, 50860, 1170466, 314862

Offset: 1

Hugo Pfoertner, Sep 08 2017

			A configuration of 6 picked points from a 7 X 7 grid with a central (point) symmetry w.r.t. point #, but no line (mirror) symmetry and thus only contributing to T(7,6)=a(27), but not to A291718(27), would be:
  o o o X o o o
  o o o o o o o
  o o o o X o o
  o X # X o o o
  X o o o o o o
  o o o o o o o
  o X o o o o o
.
Triangle begins:
   1;
   4,    6;
   9,   36,   8;
  16,  120,  24,   168;
  25,  300,  72,   714,  178;
  36,  630, 144,  2273,  464,  6576;
  49, 1176, 288,  5932, 1476, 24288,  6404;
  64, 2016, 480, 13536, 3040, 74560, 15680, 341320;

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A090642, A098485, A098487, A291716, A291718, A292152, A292153, A292154, A292155, A292156.

decentralize[v_] := 2*Total[v] - Last[v];
T[n_, k_] := decentralize[ Table[ decentralize[ Table[ If[EvenQ[k] || OddQ[a*b], Binomial[ Quotient[a*b, 2], Quotient[k, 2]], 0], {b, 1, n}]], {a, 1, n}]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

decentralize(v) = 2*vecsum(v) - v[length(v)];
T(n,k) = decentralize(vector(n, a, decentralize(vector(n, b, if(k%2==0||a*b%2==1, binomial(a*b\2, k\2))))));
for(n=1,10, for(k=1,n, print1(T(n,k), ", ")); print); \\ Andrew Howroyd, Sep 16 2017

Terms a(37) and beyond from Andrew Howroyd, Sep 16 2017

A291716 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the center of gravity of the k picked positions coincides with one of the picked positions.

Original entry on oeis.org

1, 4, 0, 9, 0, 8, 16, 0, 24, 44, 25, 0, 72, 176, 610, 36, 0, 144, 660, 2996, 12092, 49, 0, 288, 1788, 11492, 64648, 323940, 64, 0, 480, 4116, 35676, 269924, 1811696, 10866196

Offset: 1

Hugo Pfoertner, Sep 08 2017

			T(3,3) = a(6) = 8 because there are the following 8 ways to pick 3 positions such that one of them is the center of gravity of the other two.
  XXX...ooo...ooo...Xoo...oXo...ooX...Xoo...ooX
  ooo...XXX...ooo...Xoo...oXo...ooX...oXo...oXo
  ooo...ooo...XXX...Xoo...oXo...ooX...ooX...Xoo
.
An example of one of the T(4,4)=a(10)=44 "balanced" configurations is
  x.o.o.x
  o.o.X.o
  o.o.o.o
  o.o.o.x
X is at the center of gravity of the 3 other picked positions x.
.
Triangle begins:
   1;
   4, 0;
   9, 0,   8;
  16, 0,  24,   44;
  25, 0,  72,  176,   610;
  36, 0, 144,  660,  2996,  12092;
  49, 0, 288, 1788, 11492,  64648,  323940;
  64, 0, 480, 4116, 35676, 269924, 1811696, 10866196;

Cf. A090642, A098485, A098487, A291717, A291718, A292152, A292153, A292154, A292155, A292156.

A291718 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a line symmetry.

Original entry on oeis.org

1, 4, 6, 8, 36, 44, 16, 120, 192, 276, 25, 300, 596, 1130, 2010, 36, 630, 1436, 3321, 6880, 16400, 49, 1176, 3024, 8272, 20600, 57564, 120940, 64, 2016, 5568, 17528, 49184, 159784, 380344, 1075344

Offset: 1

Hugo Pfoertner, Sep 08 2017

			A configuration of 6 picked points from a 7 X 7 grid with a line (mirror) symmetry w.r.t. the line indicated by +++, and no point symmetry would be:
  o o o o o o o
  + X o X o o o
  X + o o o X o
  o o + o o o o
  X o o + o o o
  o o o o + o o
  o X o o o + o
So it would not contribute to the count of central symmetric configurations in A291717(27).
.
A configuration
  o o + o o o o
  o o + o o o o
  X o + o X o o
  + X # X + + +
  X o + o X o o
  o o + o o o o
  o o + o o o o
would contribute both to a(27) and to A291717(27), because besides being mirror symmetric w.r.t. the lines indicated by +++, it has also a central symmetry w.r.t the point indicated by #.
.
Triangle begins:
   1;
   4,    6;
   9,   36,   44;
  16,  120,  192,   276;
  25,  300,  596,  1130,  2010;
  36,  630, 1436,  3321,  6880,  16400;
  49, 1176, 3024,  8272, 20600,  57564, 120940;
  64, 2016, 5568, 17528, 49184, 159784, 380344, 1075344;

Cf. A090642, A098485, A098487, A291716, A291717, A292152, A292153, A292154, A292155, A292156.

A292152 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions don't have any symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 40, 0, 0, 368, 1432, 0, 0, 1704, 10992, 50992, 0, 0, 5704, 53784, 369776, 1925464, 0, 0, 15400, 198696, 1885128, 13903624, 85773968, 0, 0, 36096, 606264, 7572896, 74743584, 620821688, 4424756040

Offset: 1

Hugo Pfoertner, Sep 17 2017

			The triangle begins:
  0;
  0,  0;
  0,  0,    40;
  0,  0,   368,   1432;
  0,  0,  1704,  10992,  50992;
  0,  0,  5704,  53784,  369776,  1925464;
  0,  0, 15400, 198696, 1885128, 13903624, 85773968;
.
The following configuration of 6 picked points from a 7X7 grid is one of the T(7,6)=a(28)=13903624 configurations without symmetry. It is of some historical interest, because when it was drawn in Germany's "Lotto 6 aus 49", there was only one person with a winning bet receiving a payout of 22 million DM (Deutsche Mark).
.
  o o o o o o o
  o o o o o o o
  o o o o X o o
  o o X o o o o
  o o o o o o X
  X X X o o o o
  o o o o o o o

Walter Krämer, Denkste! Trugschlüsse aus der Welt der Zahlen und des Zufalls. Campus Verlag, Frankfurt/Main, 1996. Chapter 4, pp. 71-82.

Cf. A090642, A098485, A098487, A291716, A291717, A291718, A292153, A292154, A292155, A292156.

a(n) = A090642(n) - A292153(n).

A292153 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have a point symmetry or a line symmetry.

Original entry on oeis.org

1, 4, 6, 9, 36, 44, 16, 120, 192, 388, 25, 300, 596, 1658, 2138, 36, 630, 1436, 5121, 7216, 22328, 49, 1176, 3024, 13180, 21756, 80192, 126616, 64, 2016, 5568, 29112, 51616, 230784, 394504, 1409328

Offset: 1

Hugo Pfoertner, Sep 17 2017

The "or" is inclusive, i.e. configurations that have both types of symmetry simultaneously (counted separately in A292154) are included.

			The triangle begins:
   1;
   4,    6;
   9,   36,   44;
  16,  120,  192,   388;
  25,  300,  596,  1658,  2138;
  36,  630, 1436,  5121,  7216, 22328;
  49, 1176, 3024, 13180, 21756, 80192, 126616;
.
The following configuration is one of the T(4,3)=a(9)=192 symmetric configurations of 3 points picked from a 4 X 4 grid. It has both types of symmetry.
  0 0 0 0
  X 0 0 0
  0 X 0 0
  0 0 X 0

Cf. A090642, A098485, A098487, A291716, A291717, A291718, A292152, A292154, A292155, A292156.

a(n) = A090642(n) - A292152(n) = A292154(n) + A292155(n) + A292156(n).

A292154 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have both a point symmetry and a line symmetry.

Original entry on oeis.org

1, 4, 6, 9, 36, 8, 16, 120, 24, 56, 25, 300, 72, 186, 50, 36, 630, 144, 473, 128, 648, 49, 1176, 288, 1024, 320, 1660, 728, 64, 2016, 480, 1952, 608, 3560, 1520, 7326

Offset: 1

Hugo Pfoertner, Sep 17 2017

			The triangle begins:
   1;
   4,    6;
   9,   36,   8;
  16,  120,  24,   56;
  25,  300,  72,  186,  50;
  36,  630, 144,  473, 128,  648;
  49, 1176, 288, 1024, 320, 1660,  728;
  64, 2016, 480, 1952, 608, 3560, 1520, 7326;
.
  o o o o o o
  X o o X o o
  o o o o o o
  X o o X o o
  o o o o o o
  X o o X o o
is one of the T(6,6)=a(21)=648 configurations with both types of symmetry.
.
  o o X o o o
  o X o o o o
  o o o X o o
  o o o o o X
  o o o o X o
  o o o o o o
is one of the T(6,5)=a(20)=128 configurations with both types of symmetry.

Cf. A090642, A098485, A098487, A291716, A291717, A291718, A292152, A292153, A292155, A292156.

a(n) = A292153(n) - A292155(n) - A292156(n).
a(n) = A291717(n) - A292155(n).
a(n) = A291718(n) - A292156(n).

A292155 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have a point symmetry but no line symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 0, 0, 0, 528, 128, 0, 0, 0, 1800, 336, 5928, 0, 0, 0, 4908, 1156, 22628, 5676, 0, 0, 0, 11584, 2432, 71000, 14160, 333994

Offset: 1

Hugo Pfoertner, Sep 17 2017

			The triangle begins:
   0;
   0, 0;
   0, 0, 0;
   0, 0, 0,   112;
   0, 0, 0,   528,  128;
   0, 0, 0,  1800,  336,  5928;
   0, 0, 0,  4908, 1156, 22628,  5676;
   0, 0, 0, 11584, 2432, 71000, 14160, 333994;
.
The following configuration of 6 picked points from a 7X7 grid with a point symmetry but no line (mirror) symmetry is one of the T(7,6)=a(28)=22628 configurations with this property. It is of some historical interest, because when it was drawn in Germany's "Lotto 6 aus 49" in January 1988, there were 222 persons instead of typically 5-10 with a winning bet. They only won 31000 DM (Deutsche Mark) instead of the 1 million DM they had hoped for.
.
  o o o o o o o
  o o o o o o o
  o o o o o o o
  o o X X X o o
  o X X X o o o
  o o o o o o o
  o o o o o o o
.
The shown configuration is also in A098485(28) (graph consisting of a single component).

Walter Krämer, Denkste! Trugschlüsse aus der Welt der Zahlen und des Zufalls. Campus Verlag, Frankfurt/Main, 1996.

Cf. A090642, A098485, A098487, A291716, A291717, A291718, A292152, A292153, A292154, A292156.

a(n) = A292153(n) - A291718(n) = A291717(n) - A292154(n).

cp's OEIS Frontend

A098487 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square array such that all positions are mutually isolated. Two positions (s,t),(u,v) are considered as isolated from each other if min(abs(s-u),abs(t-v))>1.

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A348129 Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A098485 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square array such that their adjacency graph consists of a single component. Two positions (s,t), (u,v) are considered as adjacent if max(abs(s-u), abs(t-v)) <= 1.

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A291717 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry.

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A291716 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the center of gravity of the k picked positions coincides with one of the picked positions.

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A291718 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a line symmetry.

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A292152 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions don't have any symmetry.

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A292153 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have a point symmetry or a line symmetry.

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A292154 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have both a point symmetry and a line symmetry.

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A292155 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square grid such that the picked positions have a point symmetry but no line symmetry.

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