1,1

Let G(n) be the graph whose vertices are sequences of length n with entries in {1,2,3} for which adjacent terms differ by +/-1 and {s,t} is an edge if the i-th term of sequence s differs from the i-th term of sequence t by +/-1. Let A be the adjacency matrix of this graph. Then a(n) is the sum of the entries in A^(n-1). That is, a(n)is the number of paths of length n-1 in the graph G(n). Conjecture: a(n) = 2^A097063(n) for n even and 2*2^A097063(n) if n is odd.

Proof that a(n) = 2^A097063(n) for n even and 2*2^A097063(n) if n is odd, by Max Alekseyev (maxale(AT)gmail.com): Suppose that elements of the nxn matrix are colored in black and white like a chessboard. Then either all black or all white elements must equal 2. Each element of the other color can be 1 or 3 independently. For even n, the number of black and white elements is the same and equal n/2. For odd n, the number of black/white squares differs by 1. Therefore the number of nxn matrices defined in A146161 is 2*2^(n^2/2)=2^((n^2+2)/2) if n is even, and 2^((n^2+1)/2) + 2^((n^2-1)/2) = 3*2^((n^2-1)/2) if n is odd. The explicit formula for A097063 gives A097063(n)=(n^2+2)/2 for even n, and A097063(n)=(n^2-1)/2 for odd n. So the conjecture is true.

a(n) = 2^((n^2+2)/2) if n is even, and 3*2^((n^2-1)/2) if n is odd.

The a(2)=8 2 by 2 matrices with the desired property are {[[1, 2], [2, 1]], [[1, 2], [2, 3]], [[2, 1], [1, 2]], [[2, 1], [3, 2]], [[2, 3], [1, 2]], [[2, 3], [3, 2]], [[3, 2], [2, 1]], [[3, 2], [2, 3]]}. Here a matrix is represented as a list of its rows.

Sequence in context: A063859 A034183 A003216 this_sequence A019015 A000862 A005444

Adjacent sequences: A146158 A146159 A146160 this_sequence A146162 A146163 A146164

nonn

W. Edwin Clark (eclark(AT)math.usf.edu), Oct 27 2008

Extended by Max Alekseyev (maxale(AT)gmail.com), Mar 23 2009