A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.
1, 1, 3, 7, 27, 87, 409, 1657, 9089, 43833, 272947, 1515903, 10515147, 65766991, 501178937, 3473600465, 28773452321, 218310229201, 1949230218691, 16035686850327, 153281759047387, 1356791248984295, 13806215066685433, 130660110400259849, 1408621900803060705
Offset: 1
Keywords
Examples
Some solutions for n = 5: x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x x 2 x 0 x 0 x 2 x 0 x 1 x 1 x 2 x 2 x 1 0 x 2 x 1 x 3 x 1 x 0 x 2 x 3 x 0 x 0 x x 3 x 1 x 2 x 2 x 0 x 1 x 1 x 1 x 2 x 0 There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - _Andrew Howroyd_, Jun 06 2017
Links
- R. H. Hardin, Table of n, a(n) for n = 1..210
- Eric Weisstein's World of Mathematics, Independent Vertex Set
- Eric Weisstein's World of Mathematics, Vertex Cover
- Eric Weisstein's World of Mathematics, White Bishop Graph
Crossrefs
Programs
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Maple
a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e) , k=0..m))(iquo(n-1, 2, 'e')): seq(a(n), n=1..26); # Alois P. Heinz, Oct 03 2022 -
Mathematica
a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2]; Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 25 2022, after Francesca Aicardi *)
Formula
a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - Francesca Aicardi, May 28 2022
From Vaclav Kotesovec, Jul 29 2022: (Start)
a(2*k) = A020556(k).
a(2*k+1) = A094577(k). (End)
Comments