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A095986
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A card-arranging problem: number of permutations p_1, ..., p_n of 1, ..., n such that i + p_i is a square for every i.
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5
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1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 4, 3, 2, 5, 15, 21, 66, 37, 51, 144, 263, 601, 1333, 2119, 2154, 2189, 3280, 12405, 55329, 160895, 588081, 849906, 1258119, 1233262, 2478647, 4305500, 17278636, 47424179, 153686631, 396952852, 1043844982
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OFFSET
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0,15
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COMMENTS
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Gardner attributes the problem (for the case n = 13) to David L. Silverman.
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REFERENCES
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M. Gardner, Mathematical Games column, Scientific American, Nov 1974.
M. Gardner, Mathematical Games column, Scientific American, Mar 1975.
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 81.
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LINKS
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FORMULA
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a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether i+j is a square or not.
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EXAMPLE
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a(0) = 1: the empty permutation.
a(3) = 1: 321.
a(5) = 1: 32154.
a(8) = 1: 87654321.
a(9) = 1: 826543917.
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MAPLE
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b:= proc(s) option remember; (n-> `if`(n=0, 1, add(
`if`(issqr(n+j), b(s minus {j}), 0), j=s)))(nops(s))
end:
a:= n-> b({$1..n}):
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MATHEMATICA
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nmax=45; a[n_]:=Permanent[Table[If[IntegerQ[Sqrt[i+j]], 1, 0], {i, n}, {j, n}]]; Join[{1}, Array[a, nmax]] (* Stefano Spezia, Mar 03 2024 *)
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CROSSREFS
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KEYWORD
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nonn,hard,more,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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