A246559 -id:A246559 - cp's OEIS frontend

A246521 List of free polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000105 (in which case the offset is 0).

0, 1, 3, 7, 11, 15, 23, 27, 30, 75, 31, 47, 62, 79, 91, 94, 143, 181, 182, 188, 406, 1099, 63, 95, 111, 126, 159, 175, 183, 189, 190, 207, 219, 221, 222, 252, 347, 350, 378, 407, 413, 476, 504, 1103, 1115, 1118, 1227, 1244, 2127, 2229, 2230, 2236, 2292, 2451, 2454, 2460, 33867, 127

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 placed (and flipped/rotated) as to obtain the smallest possible value, which in particular implies pushing it to both borders. 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, is 2^0 + 2^1 + 2^3 + ... + 2^A000217(n-1) = A181388(n-1).
See A246533 for the variant that lists fixed polyominoes.

			Number the points of the first quadrant as follows:
   ...   ...   ...
    9 13 18 24 31 ...
    5  8 12 17 23 ...
    2  4  7 11 16 ...
    0  1  3  6 10 ...
An animal occupying squares numbered k1, ..., kN will be represented by a term a(n) = 2^k1 + ... + 2^kN, the position and orientation being chosen as to minimize this value:
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 free domino is rotated to the ".." configuration represented by 2^0 + 2^1 (since this is smaller than the ":" configuration with value 2^0 + 2^2).
The A000105(3) = 2 free triominoes are represented by 2^0 + 2^1 + 2^3 = [...] and 2^0 + 2^1 + 2^2 = [:.]. The latter value is smaller, therefore the L-shaped triomino is listed before the straight one.
From _M. F. Hasler_, Jan 25 2021: (Start)
Writing all N-ominoes on row N, the table begins:
  N | a(m .. m+k), m = 1 + Sum_{j<N} A000105(j), k = A000105(N) - 1
----+--------------------------------------------------------------
  0 | a(1) = 0 = []
  1 | a(2) = 1 = 2^0 = [.]
  2 | a(3) = 3 = 2^0 + 2^1 = [..]
  3 | a(4) = 7 = [:.], a(5) = 11 = [...]
  4 | 15 = [:..], 23 = [::], 27 = [.:.], 30 = [':.], 75 = [....]
... | ...
(End)

John Mason, Table of n, a(n) for n = 1..87147
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014

See A246533 and A246559 for lists of fixed and one-sided polyominoes.

Cf. A000105, A000217, A000225, A181388.

More terms from John Mason, Aug 29 2014

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),[[]])))))

A306287 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (1/6)(4+52^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th Y-type Hilbert Tree.

Original entry on oeis.org

1, 0, 3, 2, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 3, 3, 2, 1, 2, 2, 3, 0, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 1, 0

Offset: 1

Bradley Klee, Feb 03 2019

The Y-type Hilbert trees are a sequence of polyominoes whose edges, all but one, are segments of the Hilbert curve described by A163540. One extra edge closes a loop around the perimeter (cf. Formula). The first Y-type tree is a monomino with four edges, and the second is the Y hexomino with 14 unit edges. All deeper trees are determined by iteration of replacement rules (cf. linked image "First Six Y-type Trees"). The Y-type Hilbert trees nest along the upper half plane according to the limit-periodic ruler function A001511. Such an arrangement reconstructs the Hilbert curve everywhere away from the ground axis (cf. linked image "Limit-Periodic Construction").

			T(1,k) = 1, 0, 3, 2;
T(2,k) = 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 2.

Bradley Klee, Limit-Periodic Construction.
Bradley Klee, First Six Y-type Trees.

T-Type Trees: A306288. Cf. A163540, A001511, A246559.

HC = {L[n_ /; EvenQ[n]] :> {F[n], L[n], L[Mod[n + 1, 2]], R[n]},
   R[n_ /; OddQ[n]] :> {F[n], R[n], R[Mod[n + 1, 2]], L[n]},
   R[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], F[Mod[n + 1, 2]]},
   L[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], F[Mod[n + 1, 2]]},
   F[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], L[Mod[n + 1, 2]]},
   F[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], R[Mod[n + 1, 2]]}};
TurnMap = {F[] -> 0, L[] -> 1, R[_] -> -1};
T1ind[1] = 1; T1ind[2] = 2; T1ind[n_] := 5*T1ind[n - 1] - 4*T1ind[n - 2];
T1Vec[n_] := Append[Mod[FoldList[Plus, Flatten[Nest[# /. HC &, F[0],
        n] /. TurnMap][[T1ind[n] ;; -(T1ind[n] + 1)]]], 4], 2]
Flatten[T1Vec /@ Range[5]]

a(n,(1/6)*(4+5*2^(2*n))) = 2;

a(n,k) = A163540( (1/12)*(8+7*2^(2*n)-3*(-1)^n *2^(2*n+1))-1+k ).

A306288 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (2/3)(2+2^(2n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th T-type Hilbert Tree.

Original entry on oeis.org

3, 0, 1, 2, 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 2, 3, 3, 2, 3, 0, 3, 3, 2, 1, 2, 2, 3, 0, 3, 2, 3, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 2, 3, 3, 2, 3, 0, 3, 3, 2, 1, 2, 2, 3, 0, 3, 2, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0

Offset: 1

Bradley Klee, Feb 03 2019

The T-type Hilbert trees are a sequence of polyominoes whose edges, all but one, are segments of the Hilbert curve described by A163540. One extra edge closes a loop around the perimeter (cf. Formula). The first T-type tree is a monomino with four edges, and the second is the T pentomino with 12 unit edges. All deeper trees are determined by iteration of replacement rules (cf. linked image "First Six T-type Trees"). The T-type Hilbert trees nest along the upper half plane according to the limit-periodic ruler function A001511. Such an arrangement reconstructs the Hilbert curve everywhere away from the ground axis (cf. linked image "Limit-Periodic Construction").

			T(1,k) = 3, 0, 1, 2;
T(2,k) = 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 2.

Bradley Klee, Limit-Periodic Construction.
Bradley Klee, First Six T-type Trees.

Y-Type Trees: A306287. Cf. A163540, A001511, A246559.

HC = {L[n_ /; EvenQ[n]] :> {F[n], L[n], L[Mod[n + 1, 2]], R[n]},
   R[n_ /; OddQ[n]] :> {F[n], R[n], R[Mod[n + 1, 2]], L[n]},
   R[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], F[Mod[n + 1, 2]]},
   L[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], F[Mod[n + 1, 2]]},
   F[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], L[Mod[n + 1, 2]]},
   F[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], R[Mod[n + 1, 2]]}};
TurnMap = {F[] -> 0, L[] -> 1, R[_] -> -1};
T2ind[1] = 7; T2ind[2] = 27;T2ind[n_] := 5*T2ind[n - 1] - 4*T2ind[n - 2];
T2Vec[n_] := Append[Mod[ FoldList[Plus, Flatten[Nest[# /. HC &, F[0] /. HC, n] /.
       TurnMap][[T2ind[n] ;; -(T2ind[n] + 1)]]], 4], 2]
Flatten[T2Vec/@Range[5]]

a(n,(2/3)*(2+2^(2*n))) = 2;

a(n,k) = A163540( (1/3)*(1+11*2^(2*n)+3*(-1)^n *2^(2*n+1))-1+k ).

cp's OEIS Frontend

A246521 List of free polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000105 (in which case the offset is 0).

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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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A306287 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (1/6)*(4+5*2^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th Y-type Hilbert Tree.

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A306288 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (2/3)*(2+2^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th T-type Hilbert Tree.

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A306287 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (1/6)(4+52^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th Y-type Hilbert Tree.

A306288 Irregular triangle T(n,k), 1 <= n, 1 <= k <= (2/3)(2+2^(2n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th T-type Hilbert Tree.