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 A224323 Number of arrangements of particles with no nearest neighbors on a double wedge shaped lattice of height n as shown in the example. 1
 1, 2, 9, 91, 2076, 106941, 12443305, 3270757032, 1942218575613, 2605498099029291, 7896382316678271480, 54064349180204191478385, 836256337417598861881531177, 29222244645016423977808908794944, 2306924899814845893307925997931857585, 411432963638019060492077250560082497198811 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS It is related to the sequence A180998 by R. H. Hardin. In other words, a(n) is the number of independent vertex sets in the graph with n^2 vertices of the double wedge shown in the example. Neighbors are just the nearest horizontal and vertical points. The corresponding sequence for a single wedge of height n is given by A180998(n+2). - Andrew Howroyd, Apr 28 2020 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..24 Konrad Engel, On the Fibonacci Number of an M x N Lattice, The Fibonacci Quarterly, 28 (1990),  pp. 72-78. Eric Weisstein's World of Mathematics, Independent Vertex Set FORMULA The sequence was generated from a product of binary rectangular matrices S(i), where initial values are S(1)= (11) with A(1) = 1 1, and subsequent                                                       1 0 values S(n+1) = (A(n)S(n)T ) where A(n+1)= A(n)S(n)T                                            S(n)0(n) Then the vector vector product V(n)V(n)T (where T denotes transpose) gives the sequence where V(n)= S(1)S(2)......S(n). The sum of the elements of V(n) gives sequence A180998. EXAMPLE Single Wedge Lattice     x          Double Wedge Lattice        x                        x x                                    x x x                      x x x                                  x x x x x             V(3)= 11 111 11111 =  (5 4 3 5 4)                      101 10110                          11011             Double Wedge   V(3)V(3)T = 91             Single Wedge   Sum(54354) = 21 CROSSREFS Cf. A180998 (case of single wedge). Sequence in context: A059477 A136553 A266293 * A240650 A011804 A292467 Adjacent sequences:  A224320 A224321 A224322 * A224324 A224325 A224326 KEYWORD nonn AUTHOR Roger W. Haskell, Apr 03 2013 EXTENSIONS a(0)=1 prepended, a(6) corrected and terms a(10) and beyond from Andrew Howroyd, Apr 28 2020 STATUS approved

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Last modified September 24 14:46 EDT 2020. Contains 337321 sequences. (Running on oeis4.)