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 A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's. 10
 0, 1, 6, 17, 43, 84, 159, 262, 426, 635, 940, 1311, 1821, 2422, 3213, 4124, 5284, 6597, 8226, 10045, 12255, 14696, 17611, 20802, 24558, 28639, 33384, 38507, 44401, 50730, 57945, 65656, 74376, 83657, 94078, 105129, 117459, 130492, 144951, 160190, 177010 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also: Number of non-equivalent ways to place two non-attacking wazirs on an n X n board. Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood). This sequence counts equivalence classes induced by the dihedral group D_4. If equivalent matrices are distinguished, the number of matrices is A172225(n). LINKS Heinrich Ludwig, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1) FORMULA a(n) = (n^4 + 2*n^2 - 4*n)/16 if n is even; a(n) = (n^4 + 4*n^2 - 8*n + 3)/16 if n is odd. G.f.: x * (1 + x + x^2)*(1 + 3*x - x^2 + x^3) / ((1 + x)^3*(1 - x)^5). - Bruno Berselli, Nov 28 2013 EXAMPLE There are a(3) = 6 non-equivalent 3 X 3 matrices with two nonadjacent 1's (and no other 1's):   [1 0 0]    [0 1 0]    [1 0 0]    [0 1 0]    [1 0 1]    [1 0 0]   |0 0 0|    |0 0 0|    |0 1 0|    |1 0 0|    |0 0 0|    |0 0 1|   [0 0 1]    [0 1 0]    [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0] PROG (PARI) x='x+O('x^99); concat(0, Vec(x*(1+x+x^2)*(1+3*x-x^2+x^3)/((1+x)^3*(1-x)^5))) \\ Altug Alkan, Mar 14 2018 CROSSREFS Cf. A232568, A232569, A239576, A201511, A172225. Sequence in context: A047861 A171507 A099858 * A062020 A066183 A262297 Adjacent sequences:  A232564 A232565 A232566 * A232568 A232569 A232570 KEYWORD nonn,easy,changed AUTHOR Heinrich Ludwig, Nov 26 2013 STATUS approved

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Last modified March 19 16:29 EDT 2018. Contains 300866 sequences. (Running on oeis4.)