A056881
Number of n-celled diagonally symmetric polyominoes without holes.
Original entry on oeis.org
0, 0, 1, 0, 2, 2, 6, 5, 21, 18, 70, 61, 239, 206, 828, 711, 2903, 2485, 10314, 8791, 37031, 31451, 134147, 113566, 489672, 413419, 1799084, 1515282, 6647009, 5586834, 24677902, 20703840, 92009606, 77067674, 344336294, 288001040, 1292929699, 1079999410, 4869184237, 4062534430
Offset: 1
A056882
Number of n-celled polyominoes without holes, symmetric about diagonal 2.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 1, 2, 3, 4, 4, 6, 7, 12, 12, 17, 18, 33, 34, 51, 52, 93, 100, 154, 152, 276, 300, 474, 460, 846, 929, 1491, 1431, 2667, 2956
Offset: 1
A056883
Number of n-celled rotationally symmetric polyominoes without holes.
Original entry on oeis.org
0, 0, 0, 1, 1, 5, 4, 18, 19, 72, 72, 268, 276, 1002, 1043, 3733, 3926, 14000, 14810, 52658, 55987, 199020, 212364, 754826, 807861, 2872926, 3082202, 10966333, 11789248, 41970489, 45198317, 160996986, 173640109, 618839771, 668316169, 2383000591, 2576516523, 9191269314, 9947949563, 35502661446
Offset: 1
A259087
Number of n-celled polyominoes which are of rectangular but not square type.
Original entry on oeis.org
0, 1, 1, 4, 6, 28, 83, 289, 1030, 3860, 14154, 53222, 202001, 771255, 2945803, 11304541, 43564276, 168413249, 652292925, 2531493604, 9845752518, 38372748112
Offset: 1
- T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21. The 15th entry is not included because their value for A000105(15) was incorrect.
A259090
Number of symmetrical n-celled polyominoes.
Original entry on oeis.org
1, 1, 2, 4, 7, 15, 24, 53, 89, 194, 323, 722, 1197, 2706, 4465, 10229, 16814, 38900, 63721, 148808, 243006, 571937, 931305, 2207561, 3584959, 8551309, 13851410, 33229843, 53694326, 129485500, 208738348, 505796115, 813532119, 1980023840, 3177764415
Offset: 1
- John Mason, Table of n, a(n) for n = 1..50 (This version corrects erroneous values from a(44) onwards in previous version)
- T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21. The 15th entry is not included because their value for A000105(15) was incorrect.
-
A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt", "Table"], {, }][[All, 2]];
a[n_] := A000105[[n + 1]] - A006749[[n]];
Array[a, 28] (* Jean-François Alcover, Aug 27 2019 *)
A308300
T(n,k) is the number of simply connected square animals with n cells and k internal vertices (0 <= k <= A083479(n)), triangle read by rows.
Original entry on oeis.org
1, 1, 2, 4, 1, 11, 1, 27, 7, 1, 82, 21, 4, 250, 90, 21, 2, 815, 334, 89, 9, 1, 2685, 1311, 391, 67, 6, 9072, 4978, 1674, 324, 45, 1, 30889, 19030, 7089, 1630, 275, 23, 1, 106290, 72082, 29433, 7629, 1498, 174, 11
Offset: 1
The table starts:
1;
1;
2;
4, 1;
11, 1;
27, 7, 1;
82, 21, 4;
250, 90, 21, 2;
815, 334, 89, 9, 1;
2685, 1311, 391, 67, 6;
9072, 4978, 1674, 324, 45, 1;
30889, 19030, 7089, 1630, 275, 23, 1;
106290, 72082, 29433, 7629, 1498, 174, 11;
...
A359520
Number of free holey polyominoes of n cells with simply-connected interiors.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 5, 35, 179, 909, 4285, 19702, 88040, 386826, 1674852, 7174436, 30462009
Offset: 1
a(7) = 1 because of the holey heptomino.
A056879
Number of n-celled axially symmetric polyominoes without holes.
Original entry on oeis.org
0, 0, 0, 1, 2, 6, 9, 23, 37, 87, 138, 313, 510, 1130, 1859, 4070, 6786, 14750, 24811, 53715, 91096, 196769, 335815, 724470, 1243175, 2680212, 4619999, 9957652, 17231447, 37137177, 64480255, 138978282, 242011375, 521706359, 910817416, 1963885190, 3436463468, 7411495499, 12995261895, 28034824487
Offset: 1
A056884
Number of asymmetric n-celled polyominoes without holes.
Original entry on oeis.org
0, 0, 0, 1, 5, 20, 84, 311, 1165, 4276, 15802, 58280, 216070, 803116, 2997360, 11221443, 42147842, 158749808, 599513836, 2269390757, 8609258335, 32725209938, 124621150623, 475367241264, 1816100804241, 6948222137523, 26618662003342, 102102765921611, 392095326233619, 1507357249572269, 5800716924062537, 22344021033377558
Offset: 1
A057051
Number of polyominoes of 2n-1 cells that span an n X n square.
Original entry on oeis.org
1, 1, 6, 18, 73, 255, 950, 3473, 13006, 48840, 185353, 706404, 2706608, 10404625, 40126430, 155133811, 601119492, 2333671638, 9075290555, 35345525798, 137847145330, 538258922839, 2104101413400, 8233434921693, 32247613423563, 126410623214720, 495918571702575
Offset: 1
- D. E. Knuth, Animals in a cage, Problem 10875, Amer. Math. Monthly, 110 (March 2003), 243-245.
- R. J. Mathar, Corrigendum to "Polyomino Enumeration Results (Parkin et al, SIAM Fall Meeting 1967)" viXra:1905.0474 (2019)
- R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy) page 9 (incorrect at n=15).
-
A057051 := proc(n) if n mod 2 = 0 then binomial(2*n-2,n-1)+2^(n-2)-(3*n^2-2*n+8)/8; else binomial(2*n-2,n-1)+2^(n-2)-(3*n^2-4*n+9)/8+(1/2)*binomial(n-1,(n-1)/2); end if; end proc;
-
f[n_] := If[EvenQ[n], Binomial[2n-2, n-1] + 2^(n-2) - (3n^2-2n+8)/8, Binomial[2n-2, n-1] + 2^(n-2) - (3n^2-4n+9)/8 + (1/2) Binomial[n-1, (n-1)/2]]; Table[f[n], {n, 1, 27}] (* Jean-François Alcover, Mar 18 2017, translated from Maple *)
Comments