A246521 -id:A246521 - cp's OEIS frontend

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 1, 3, 5, 0, 0, 0, 0, 7, 6, 3, 5, 0, 0, 0, 0, 1, 5, 3, 4, 5, 0, 0, 0, 0, 8, 6, 1, 6, 1, 3, 0, 0, 0, 0, 7, 2, 1, 2, 0, 7, 8, 0, 0, 0, 0, 2, 5, 7, 9, 7, 9, 1, 1, 0, 0, 0, 0, 2, 5, 4, 4, 5, 4, 3, 6, 1, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the number on the n-th row of A368660 divided by A335573(n+1). See A368660 for details.

Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2.

			Array begins:
  1.00000000000000000000... (monomino)
  0.50000000000000000000... (domino)
  0.14317187227209462175... (L tromino)
  0.21365625545581075649... (I tromino)
  0.05331174468766310877... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.05107523273680265528... (T tetromino)
  0.03794485956843370668... (S tetromino)
  0.08139812221208792734... (I tetromino)
  0.01652391644265825925... (P pentomino)
  0.01709341200261444870... (V pentomino)
  0.00933365290110550590... (W pentomino)
  0.01825698429438352158... (L pentomino)
  0.01973313069852314774... (Y pentomino)
  0.01316184592639931744... (N pentomino)
  0.01069856796007681265... (U pentomino)
  0.02067501830899727807... (T pentomino)
  0.01358243200363682514... (F pentomino)
  0.01232428737930631004... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.02831865405554939733... (I pentomino)
  ...

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Index entries for sequences related to polyominoes.

Cf. A000105, A001168, A130866, A246521, A335573, A368660 (free polyominoes), A368864, A368865.

Corresponding sequences for internal diffusion-limited aggregation: A368392, A368393.

A367439 a(n) is the degree of the polyomino with binary code A246521(n+1) in the polyomino graph PG(n) defined in A367435.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 4, 3, 2, 10, 8, 3, 9, 10, 9, 8, 9, 10, 8, 4, 2, 15, 28, 15, 12, 12, 10, 17, 14, 19, 20, 15, 14, 15, 13, 18, 20, 9, 14, 13, 17, 4, 12, 16, 18, 11, 9, 10, 15, 22, 19, 10, 19, 14, 16, 3, 36, 36, 35, 31, 28, 30, 36, 22, 29, 37, 16, 11, 28, 13, 24

Offset: 1

Pontus von Brömssen, Nov 18 2023

Number of free polyominoes that can be made from the polyomino with binary code A246521(n+1) by moving one of its cells (not counting itself), where the intermediate (the set of cells remaining when the cell to be moved is detached) is required to be a (connected) polyomino.

Can be read as an irregular triangle, whose m-th row contains A000105(m) terms, m >= 1.

			As an irregular triangle:
   0;
   0;
   1, 1;
   4, 3, 4, 3,  2;
  10, 8, 3, 9, 10, 9, 8, 9, 10, 8, 4, 2;
  ...
For n = 8, A246521(8+1) = 30 is the binary code of the S-tetromino. By moving one cell of the S-tetromino, we can obtain the L, O, and T tetrominoes (but not the I tetromino), so a(8) = 3.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A367126, A367435, A367437 (row maxima), A367438, A367443.

a(n) <= A367126(n).

A365963 Let I(n) be the moment of inertia of the polyomino with binary code A246521(n+1) about an axis through its center of mass perpendicular to the plane of the polyomino, the polyomino having a unit point mass in the center of each of its cells. a(n) is I(n) times the number of cells of the polyomino.

Original entry on oeis.org

0, 1, 4, 6, 14, 8, 11, 12, 20, 20, 32, 28, 38, 30, 32, 26, 26, 24, 30, 20, 50, 40, 49, 69, 61, 33, 49, 37, 46, 41, 52, 41, 53, 42, 61, 53, 52, 61, 34, 41, 50, 57, 85, 70, 73, 65, 69, 65, 60, 53, 56, 69, 49, 44, 45, 105, 82, 58, 64, 88, 86, 76, 74, 94, 86, 82

Offset: 1

Pontus von Brömssen, Sep 23 2023

If the cells have a uniform density of 1 instead of point masses in the centers, the moment of inertia is I(n) + k/6 = a(n)/k + k/6, where k is the number of cells.

			As an irregular triangle:
  0;
  1;
  4, 6;
  14, 8, 11, 12, 20;
  20, 32, 28, 38, 30, 32, 26, 26, 24, 30, 20, 50;
  ...
The five tetrominoes have moments of inertia 7/2, 2, 11/4, 3, 5 (in the order they appear in A246521). Multiplying these numbers by 4, we obtain the 4th row.
The last term of the k-th row of the irregular triangle corresponds to the straight k-omino, whose moment of inertia is k*(k^2-1)/12, so the last term of the k-th row is k^2*(k^2-1)/12 = A002415(k). (This ought to be the largest term of the k-th row.)

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Index entries for sequences related to moment of inertia.
Index entries for sequences related to polyominoes.

Cf. A000105 (row lengths), A002415, A246521, A365964 (row minima).

If the centers of the cells of the polyomino have coordinates (x_i,y_i), 1 <= i <= k, its moment of inertia is Sum_{i=1..k} x_i^2+y_i^2 - (Sum_{i=1..k} x_i)^2/k - (Sum_{i=1..k} y_i)^2/k.

A364927 List of free polyplets in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

Original entry on oeis.org

1, 3, 6, 7, 11, 14, 25, 56, 15, 23, 27, 29, 30, 46, 57, 58, 75, 78, 89, 92, 120, 166, 177, 178, 198, 209, 240, 390, 452, 960, 31, 47, 59, 62, 79, 91, 93, 94, 110, 121, 122, 124, 143, 167, 174, 179, 181, 182, 185, 186, 188, 199, 206, 211, 213, 230, 241, 242

Offset: 1

Pontus von Brömssen, Aug 13 2023

Can be read as an irregular triangle, whose n-th row contains A030222(n) terms.

			As irregular triangle:
  1;
  3,  6;
  7, 11, 14, 25, 56;
  ...
The A030222(3) = 5 3-polyplets are oriented as follows to obtain their binary codes (see A246521):
  . . .   . . .   . . .   . . .   5 . .
  2 . .   . . .   2 . .   . 4 .   . 4 .
  0 1 .   0 1 3   . 1 3   0 . 3   . . 3
This gives the binary codes 2^0+2^1+2^2 = 7, 2^0+2^1+2^3 = 11, 2^1+2^2+2^3 = 14, 2^0+2^3+2^4 = 25, and 2^3+2^4+2^5 = 56, respectively.

Pontus von Brömssen, Table of n, a(n) for n = 1..22449 (rows 1..8).

Cf. A030222, A246521.

A364928 List of free corner-connected polyominoes in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

Original entry on oeis.org

1, 6, 25, 56, 57, 198, 390, 452, 960, 454, 962, 2105, 3097, 3128, 4153, 7185, 10296, 14353, 15392, 31744, 65988, 966, 3129, 6201, 7193, 7217, 7224, 10297, 11320, 14361, 14392, 15377, 15400, 15408, 31752, 31760, 65990, 66498, 66500, 98502, 98756, 99264

Offset: 1

Pontus von Brömssen, Aug 13 2023

Corner-connected polyominoes are in one-to-one correspondence with ordinary polyominoes, but their binary codes differ and the order in which they appear here is different from that in A246521. The first size for which the order differs from A246521 is 4 (tetrominoes). Here the order of the tetrominoes is (T, S, square, L, straight), whereas in A246521 it is (L, square, T, S, straight).

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As irregular triangle:
   1;
   6;
  25,  56;
  57, 198, 390, 452, 960;
  ...
The corner-connected trominoes are oriented as follows to obtain their binary codes (see A246521):
  . . .   5 . .
  . 4 .   . 4 .
  0 . 3   . . 3
This gives the binary codes 2^0+2^3+2^4 = 25 and 2^3+2^4+2^5 = 56, respectively.
Similarly, for the corner-connected tetrominoes, the orientations
  . . . .   . . . .   . . . .   . . . .   9 . . .
  5 . . .   . . . .   . 8 . .   . 8 . .   . 8 . .
  . 4 . .   2 . 7 .   2 . 7 .   2 . 7 .   . . 7 .
  0 . 3 .   . 1 . 6   . 1 . .   . . . 6   . . . 6
give the binary codes 57, 198, 390, 452, 960, respectively.

Cf. A000105, A246521.

A380597 Smallest side length of a square board on which Harary's generalized tic-tac-toe (or animal tic-tac-toe) for the free polyomino with binary code A246521(n+1) is a first-player win, or 0 if it is a draw for all board sizes.

Original entry on oeis.org

1, 2, 3, 4, 4, 0, 5, 3, 7, 0, 0, 0, 7, 7, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 27 2025

The only unknown value is a(45), corresponding to the "long N" hexomino. It has been suggested that a(45) = 15 and A380598(45) = 13.
Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.
The number of free polyominoes of size k = 1, 2, ... for which the game is a first-player win is 1, 1, 2, 4, 3, x, 0, 0, ..., where x is 0 or 1 and all terms after x are 0.

			As an irregular triangle:
  1;
  2;
  3, 4;
  4, 0, 5, 3, 7;
  0, 0, 0, 7, 7, 6, 0, 0, 0, 0, 0, 0;
  ...
For n = 9, the polyomino with binary code A246521(9+1) = 75 is the straight tetromino. Generalized tic-tac-toe for this polyomino (i.e., 4 cells in a row, horizontally or vertically, are needed to win) is a draw for square boards of side length less than 7, but on a 7 X 7 board the first player can force a win in at most 8 moves, so a(9) = 7.

Brady Haran and Sophie Maclean, Snakey Hexomino, Numberphile video, 2025.
Wikipedia, Harary's generalized tic-tac-toe.
James Yolkowski, Tic-tac-toe.
Index entries for sequences related to polyominoes.
Index entries for sequences related to tic-tac-toe.

Cf. A000105, A246521, A380598.

a(n) = 0 for all n >= 46.

A380598 Number of moves required for the first player to win Harary's generalized tic-tac-toe (or animal tic-tac-toe) for the free polyomino with binary code A246521(n+1) on a square board of side length A380597(n), or 0 if it is a draw for all board sizes.

Original entry on oeis.org

1, 2, 3, 3, 4, 0, 4, 5, 8, 0, 0, 0, 10, 9, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 27 2025

The only unknown value is a(45), corresponding to the "long N" hexomino. It has been suggested that a(45) = 13 and A380597(45) = 15.

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As an irregular triangle:
  1;
  2;
  3, 3;
  4, 0, 4,  5, 8;
  0, 0, 0, 10, 9, 6, 0, 0, 0, 0, 0, 0;
  ...
For n = 9, the polyomino with binary code A246521(9+1) = 75 is the straight tetromino. Generalized tic-tac-toe for this polyomino (i.e., 4 cells in a row, horizontally or vertically, are needed to win) is a draw for square boards of side length less than 7, but on a 7 X 7 board the first player can force a win in at most 8 moves, so a(9) = 8.

Brady Haran and Sophie Maclean, Snakey Hexomino, Numberphile video, 2025.
Wikipedia, Harary's generalized tic-tac-toe.
James Yolkowski, Tic-tac-toe.
Index entries for sequences related to polyominoes.
Index entries for sequences related to tic-tac-toe.

Cf. A000105, A246521, A380597.

a(n) = 0 for all n >= 46.

A246559 List of one-sided polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000988.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 27, 30, 39, 54, 75, 31, 47, 55, 62, 79, 91, 94, 143, 181, 182, 188, 203, 286, 314, 406, 551, 566, 1099, 63, 95, 111, 126, 159, 175, 183, 189, 190, 207, 219, 221, 222, 252, 287, 303, 315, 318, 347, 350, 378, 407, 413, 476, 504

Offset: 1

M. F. Hasler, Aug 29 2014

nonn

The binary coding (as suggested in a post to the SeqFan list by F. T. Adams-Watters) is obtained by summing the powers of 2 corresponding to the numbers covered by the polyomino, when the points of the quarter-plane are numbered by antidiagonals, and the animal is pushed to both borders as to obtain the smallest possible value. See example for further details.

The smallest value for an n-omino is the sum 2^0+...+2^(n-1) = 2^n-1 = A000225(n), and the largest value, obtained for the straight n-omino (in x direction), is 2^0+2^1+2^3+...+2^A000217(n-1) = A181388(n-1).

			Number the points of the first quadrant as follows:
...
9 ...
5 8 ...
2 4 7 ...
0 1 3 6 10 ...
The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
The dominos ".." and ":" would be represented by 2^0+2^1 = 3 and 2^0+2^2 = 5. Since they are equivalent up to rotation, only 3 = a(3) is listed.
The A000988(3) = 2 one-sided trominoes are represented by 2^0+2^1+2^3 = 11 (...) and 2^0+2^1+2^2 = 7 (:.). Again these values are listed in increasing size as a(4) and a(5).

John Mason, Table of n, a(n) for n = 1..46575
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246521 and A246533 for enumeration of free and fixed polyominoes.

rot(P,T=[0,1;-1,0])=P=Set(apply(x->x*T,P));apply(x->x-[P[1][1],0],P)
onesided(L,N=apply(p2n,L))={ local(L=L, R=apply(P->setsearch(L,rot(P)),L), cleanup(i)=my(m=N[i]); while(m!=N[i=R[i]], if( m>N[i], m=N[i], L[i]=0))); for(i=1,#L, L[i] && cleanup(i));if(#L>1,select(P->P,L),L)}
for(i=0,5,print(Set(apply(p2n,onesided(L=if(i,grow(L),[[]])))))) \\ see A246533 for grow() and p2n()

A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 19, 21, 22, 37, 15, 23, 27, 30, 39, 53, 54, 75, 139, 147, 149, 150, 156, 275, 277, 278, 293, 306, 549, 31, 47, 55, 62, 79, 91, 94, 143, 151, 155, 157, 158, 181, 182, 188, 203, 220, 279, 283, 286, 295, 307, 309, 310, 314, 403, 405, 406, 412, 434, 440

Offset: 1

M. F. Hasler, Aug 28 2014

nonn

The binary coding (as suggested in a post to the SeqFan list by F. T. Adams-Watters) is obtained by summing the powers of 2 corresponding to the numbers covered by the polyomino, when the points of the quarter-plane are numbered by antidiagonals, and the animal is pushed to both borders as to obtain the smallest possible value. See example for further details.

The smallest value for an n-omino is the sum 2^0+...+2^(n-1) = 2^n-1 = A000225(n), and the largest value, obtained for the straight n-omino in y direction, is 2^0+2^2+2^5+...+2^(A000217(n)-1) = A246534(n).

			Number the points of the first quadrant as follows:
...
9 ...
5 8 ...
2 4 7 ...
0 1 3 6 10 ...
The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
The two fixed dominos are ".." and ":", represented by 2^0+2^1 = 3 = a(3) and 2^0+2^2 = 5 = a(4).
The A001168(3) = 6 fixed trominoes are represented by 2^0+2^1+2^3 = 11 (...), 2^0+2^1+2^2 = 7 (:.), 2^0+2^1+2^4 =19 (.:), ..., 2^0+2^2+2^5 = 37; again these 6 values are listed in increasing size as a(5), ..., a(10).

John Mason, Table of n, a(n) for n = 1..50149
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246521 and A246559 for enumeration of free and one-sided polyominoes.

grow(L,N=[],D=[[1,0],[0,1],[-1,0],[0,-1]])={ for(i=1,#L,for(j=1,#P=L[i],for(k=1,#P,for(d=1,#D, vecmin(P[k]+D[d])<0 && P-=vector(#P,i,D[d])/*shift if needed*/; !setsearch(P,P[k]+D[d]) && N=setunion([setunion([P[k]+D[d]],P)],N); P!=L[i] && P+=vector(#P,i,D[d])/*undo...*/))));if(N,N,[[[0,0]]])}
p2n(P)=sum(i=1,#P,2^(P[i][2]+A000217(P[i][1]+P[i][2])))
for(i=0,5,print(vecsort(apply(p2n,L=if(i,grow(L),[[]])))))

A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

Original entry on oeis.org

1, 3, 7, 19, 15, 23, 39, 43, 51, 54, 1043, 31, 47, 55, 59, 87, 118, 173, 179, 182, 199, 230, 1047, 1075, 1078, 2071, 2075, 2149, 2150, 2164, 2214, 2218, 6182, 1049619, 63, 95, 119, 175, 183, 190, 207, 215, 231, 237, 238, 246, 423, 430, 438, 1055, 1079, 1083

Offset: 1

Pontus von Brömssen, Aug 23 2023

The binary code used here is a straight-forward generalization of the binary code in A246521 to d > 2 dimensions. Order the d-tuples of nonnegative integers, first according to their sum, then colexicographically. (For the purposes of this definition, the result will be the same if we use lexicographic order instead.) Label the d-tuples 0, 1, 2, ... in this order. (For d = 3, this is the ordering of triples given by A144625.) Given a d-dimensional polyomino (represented as a finite set of integer d-tuples), consider all the d!*2^d ways of rotating/reflecting it. Translate each such rotation/reflection so that the minimum coordinate is 0 in each dimension, and add the powers of 2 with exponents equal to the labels of the d-tuples of the translation. The binary code of the polyomino (or any finite set of d-tuples) is the minimum of those sums.

Can be read as an irregular triangle, whose n-th row contains A038119(n) terms.

			Consider the pentacube consisting of a straight tricube with two monocubes attached to two adjacent faces of its middle cube. The following table shows the first few triples (with their ordinal number in front), with those triples appearing in the orientation of the pentacube that minimizes the binary code marked with an "X":
  0. 000 X
  1. 100 X
  2. 010
  3. 001
  4. 200 X
  5. 110 X
  6. 020
  7. 101 X
  8. 011
  9. 002
Consequently, the binary code of this pentacube is 2^0+2^1+2^4+2^5+2^7 = 179 = a(19).
As an irregular triangle:
  1;
  3;
  7, 19;
  15, 23, 39, 43, 51, 54, 1043;
  ...

Index entries for sequences related to polyominoes.

Cf. A038119, A144625, A246521 (2 dimensions), A365140 (4 dimensions), A365141 (5 dimensions).

cp's OEIS Frontend

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

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A367439 a(n) is the degree of the polyomino with binary code A246521(n+1) in the polyomino graph PG(n) defined in A367435.

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A364927 List of free polyplets in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

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A364928 List of free corner-connected polyominoes in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

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A380597 Smallest side length of a square board on which Harary's generalized tic-tac-toe (or animal tic-tac-toe) for the free polyomino with binary code A246521(n+1) is a first-player win, or 0 if it is a draw for all board sizes.

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A380598 Number of moves required for the first player to win Harary's generalized tic-tac-toe (or animal tic-tac-toe) for the free polyomino with binary code A246521(n+1) on a square board of side length A380597(n), or 0 if it is a draw for all board sizes.

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A246559 List of one-sided polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000988.

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A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

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A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.