A165632 -id:A165632 - cp's OEIS frontend

A165633 Number of tatami-free rooms of given size A165632(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 4, 2, 1, 1, 1

Offset: 1

M. F. Hasler, Sep 26 2009

nonn

Number of rectangles of size A165632(n) which cannot be tiled with tatamis of size 1x2 such that not more than 3 tatamis meet at any point.

			a(1)=1 because the rectangle of size 7x10 is the only one of size 70 that cannot be filled with 2x1 tiles without having 4 tiles meet in some point.
a(237)=5 because there are 5 different rectangles of size A165632(237)=1320 which cannot be tiled in the given way.

Project Euler, Problem 256: Tatami-Free Rooms, Sept. 2009.

Cf. A068920.

A165633 = #{ {r,c} | rc = A165632(n) }.

A165764 Smallest size of which there are n tatami-free rooms.

Original entry on oeis.org

70, 198, 336, 504, 1320, 1440, 3696, 3360, 5040, 8400, 6720, 10080, 16632, 16800, 18480, 20160, 15120, 33264, 37800, 30240, 45360, 73920, 60480, 65520, 85680, 55440, 124740, 142560, 138600, 151200, 131040, 180180, 257040, 110880, 166320

Offset: 1

M. F. Hasler, Sep 26 2009

nonn

A tatami-free room is a rectangle of even size that allows no 1x2 domino tiling satisfying the tatami rule, i.e. such that there is no point in which 4 tiles meet.

a(n)=A165632(A165765(n)) where A165765(n) is the least index for which A165633(A165765(n))=n.

			The smallest tatami-free room is of size 7x10, and all other rectangles of this size allow for a tatami tiling, thus a(1) = 70.
a(5)=1320 is the smallest size of which there are exactly 5 tatami-free rooms, namely 20x66, 22x60, 24x55, 30x44 and 33x40.

Project Euler, Problem 256: Tatami-Free Rooms, Sept. 2009.

A165764(n) = A165632(A165765(n)) = min { r*c in 2Z | #{{r,c} | A068920(r,c)=0 } = n }

A165761 Indices of records in A165633.

Original entry on oeis.org

1, 14, 36, 68, 237, 263, 709, 1111, 1520, 2350, 3605, 7437, 11325, 13921, 28337, 42896, 57483, 85499, 94206, 108922, 145872, 190311, 286903, 383819, 578215, 668153, 773158

Offset: 1

M. F. Hasler, Sep 26 2009

nonn

A165632(a(n))=A165762(n) are areas for which there are more

Project Euler, Problem 256: Tatami-Free Rooms, Sept. 2009.

A165763 Records in A165633.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 18, 21, 22, 27, 33, 35, 39, 40, 43, 46, 51, 59, 63, 70, 74, 76

Offset: 1

M. F. Hasler, Sep 26 2009

nonn

A165632(A165761(n))=A165762(n) are areas for which there are more tatami-free rooms than for any smaller size, this sequence gives that number of tatami-free rooms.

Project Euler, Problem 256: Tatami-Free Rooms, Sept. 2009.

A165763(n)=A165633(A165761(n))

A165766 Number of tatami-free rooms of size 2n.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Sep 26 2009

nonn

Number of pairs (r,c) such that r<=c, r*c=2n and A068920(r,c)=0.

A165632 lists twice the indices of nonzero terms, i.e. A165632 = { 2n | a(n)=0 }.

cp's OEIS Frontend

A165633 Number of tatami-free rooms of given size A165632(n).

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A165764 Smallest size of which there are n tatami-free rooms.

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A165761 Indices of records in A165633.

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A165763 Records in A165633.

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A165766 Number of tatami-free rooms of size 2n.

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