cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A068920 Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 4, 0, 4, 0, 1, 6, 4, 4, 6, 1, 0, 9, 0, 2, 0, 9, 0, 1, 13, 6, 3, 3, 6, 13, 1, 0, 19, 0, 3, 0, 3, 0, 19, 0, 1, 28, 10, 3, 2, 2, 3, 10, 28, 1, 0, 41, 0, 5, 0, 2, 0, 5, 0, 41, 0, 1, 60, 16, 5, 2, 2, 2, 2, 5, 16, 60, 1, 0, 88, 0, 6, 0, 1, 0, 1, 0, 6, 0, 88, 0, 1, 129, 26
Offset: 1

Views

Author

Dean Hickerson, Mar 11 2002

Keywords

Comments

Rows 2-6 are given in A068921 - A068925.

Examples

			Table begins:
  0, 1, 0, 1, 0, 1, ...
  1, 2, 3, 4, 6, 9, ...
  0, 3, 0, 4, 0, 6, ...
  1, 4, 4, 2, 3, 3, ...
  0, 6, 0, 3, 0, 2, ...
  1, 9, 6, 3, 2, 2, ...
  ...

Links

Crossrefs

Cf. A068926 for incongruent tilings, A067925 for count by area.
Cf. A068921 (row 2), A068922 (row 3), A068923 (row 4), A068924 (row 5), A068925 (row 6).

Programs

  • Mathematica
    (* See link for Mathematica programs. *)
c[r_, s_] := Which[s<0, 0, r==1, 1 - Mod[s, 2], r == 2, c1[2, s] + c2[2, s] + Boole[s == 0], OddQ[r], c[r, s] = c[r, s - r + 1] + c[r, s - r - 1] + Boole[s == 0], EvenQ[r], c[r, s] = c1[r, s] + c2[r, s] + Boole[s == 0]];
c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];
c2[r_, s_] := Which[s <= 0, 0, r == 2, c2[2, s] = c1[2, s - 2] + Boole[s == 2], EvenQ[r], c2[r, s] = c1[r, s - r + 2] + c1[r, s - r] + Boole[s == r - 2] + Boole[s == r]];
t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
A068920[n_] := Module[{x}, x = Floor[(Sqrt[8 n + 1] - 1)/2]; t[n + 1 - x (x + 1)/2, (x + 1) (x + 2)/2 - n]];
Table[A068920[n], {n, 0, 100}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

Original entry on oeis.org

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577
Offset: 1

Views

Author

R. J. Mathar, Apr 30 2016

Keywords

Comments

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

Examples

			The triangle starts in row n=1 and column m=0 as:
1,3,1;
4,18,7;
4,27,13;
2,32,32;
3,52,64,7;
3,62,133,40;
3,99,269,110,9;
5,152,437,280,48;
5,163,730,669,138,9;
6,258,1243,1318,433,48;
8,343,1823,2670,1239,154,9;
8,408,2949,5240,2849,600,48;
11,632,4577,9011,6655,1927,172,9;
13,746,6287,16184,14697,4930,777,48;
14,971,9928,28135,28805,13089,2669,190,9;
19,1394,14234,44806,58022,32176,7501,954,48;
21,1610,19501,75702,111795,70427,22344,3445,208,9;
25,2224,29785,121302,199354,157078,59859,10576,1131,48;
32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;
35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;
44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;
53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;
60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;
76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;

Crossrefs

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

Formula

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016

A068929 Number of incongruent ways to tile a 4 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

1, 3, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 11, 12, 14, 17, 20, 24, 29, 32, 41, 46, 56, 68, 78, 93, 114, 130, 161, 188, 223, 268, 318, 378, 456, 533, 646, 763, 911, 1092, 1296, 1542, 1855, 2190, 2634, 3133, 3732, 4463, 5323, 6339, 7596, 9022, 10802, 12876
Offset: 1

Views

Author

Dean Hickerson, Mar 11 2002

Keywords

Links

Crossrefs

Cf. A068923 for total number of tilings, A068926 for more info.

Formula

For n >= 20, a(n) = a(n-3) + a(n-5) + a(n-6) - a(n-9) + a(n-10) - a(n-11) - a(n-13) - a(n-15).
G.f.: x*(1-x^18+x^17+x^16+x^15+x^13-x^12-2*x^11-2*x^8-4*x^7-3*x^6-x^5-x^4+2*x^2+3*x) / ((x^5+x^3-1) * (x^10+x^6-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]

Extensions

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.
Showing 1-3 of 3 results.

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