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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)



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


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


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


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.


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


            Double Wedge   V(3)V(3)T = 91

            Single Wedge   Sum(54354) = 21


Cf. A180998 (case of single wedge).

Sequence in context: A059477 A136553 A266293 * A240650 A011804 A292467

Adjacent sequences:  A224320 A224321 A224322 * A224324 A224325 A224326




Roger W. Haskell, Apr 03 2013


a(0)=1 prepended, a(6) corrected and terms a(10) and beyond from Andrew Howroyd, Apr 28 2020



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