A143659 Number of perfect matchings of the (2n+1) X (2n+1) grid H_n with a central unit hole.

1, 2, 196, 75272, 599466256, 28838245503008, 22463213552677201984, 123818965842734619629420672, 9830731407624872890044690474098944, 6011432546485776316904414215762657381908992

Offset

0,2

Links

Table of n, a(n) for n=0..9.

Mihai Ciucu, Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole, arXiv:0710.4500 [math.CO], 2007.

Formula

a(n) = A270668(n,n). - Alois P. Heinz, Mar 21 2016

Example

The number of perfect matchings of the (2n+1) X (2n+1) grid H_n with a central unit hole does not seem to factor into a product of small primes. We have the following prime factorizations: a(1) = M(H_1) = 2. a(2) = M(H_2) = 196 = 2^2 * 7^2. a(3) = M(H_3) = 75272 = 2^3 * 97^2. a(4) = M(H_4) = 599466256 = 2^4 * 6121^2. a(5) = M(H_5) = 28838245503008 = 2^5 * 31^2 * 113^2 * 271^2. a(6) = M(H_6) = 22463213552677201984 = 2^6 * 592442159^2. a(7) = M(H_7) = 123818965842734619629420672 = 2^7 * 7417^2 * 132605129^2. a(8) = M(H_8) = 2^8 * 4481^2 * 8513^2 * 9929^2 * 16361^2.
a(9) = M(H_9) = 6011432546485776316904414215762657381908992 = 2^9 * 4639^2 * 23357676333902111^2. a(10) = M(H_10) = 49438198985375823847222358907915781467506320590324736 = 2^10 * 7^2 * 73^2 * 191^2 * 479^2 * 51151^2 * 2905610745223^2. a(11) = M(H_11) = 3302685794941188104245211026600715429110809533132141649177479168 = 2^11 * 1033^2 * 1049^2 * 1663^2 * 166151^2 * 4241286739685449^2. a(12) = M(H_12) = 2836223684393795085092141247901684583089503537241007344342697156002091831296 = 2^12 * 41^2 * 137^2 * 7057^2 * 20992575527970355281835400921^2.

Crossrefs

Main diagonal of A270668.

Sequence in context: A316901 A316888 A316890 * A282700 A256410 A174367

Adjacent sequences: A143656 A143657 A143658 * A143660 A143661 A143662

Keyword

nonn

Author

Jonathan Vos Post, Aug 28 2008

Extensions

a(0)=1 from Alois P. Heinz, Mar 21 2016

Status

approved