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A073451
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Number of essentially different ways in which the squares 1,4,9,...,n^2 can be arranged in a sequence such that all pairs of adjacent squares sum to a prime number. Rotations and reversals are counted only once.
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3
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1, 1, 1, 1, 2, 4, 0, 12, 6, 66, 156, 44, 312, 1484, 2672, 6680, 19080, 45024, 168496, 2033271, 724543, 2776536, 24598062, 26849699, 345160845, 4478968678, 5094833662, 14184530127, 29116554754, 125878922175
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OFFSET
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1,5
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COMMENTS
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Note that when the first and last numbers of an arrangement sum to a prime, then there are n rotations that are treated as one arrangement. The case n=10 exhibits the first of these rotational solutions: {1,4,9,64,49,100,81,16,25,36}. The Mathematica program uses a backtracking algorithm to count the arrangements. To print the unique arrangements, remove the comments from around the print statement.
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LINKS
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EXAMPLE
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a(5)=2 because there are two essential different arrangements: {9,4,1,16,25} and {9,4,25,16,1}.
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MATHEMATICA
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nMax=12; $RecursionLimit=500; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=PrimeQ[soln[[1]]^2+soln[[n]]^2]; If[(!circular&&soln[[1]]<soln[[n]])||(circular&&soln[[1]]==1&&soln[[2]]<=soln[[n]]), (*Print[soln^2]; *)cnt++ ], (*else append another number to the soln list*) t=soln[[lev-1]]; For[j=1, j<=Length[s[[t]]], j++, If[ !MemberQ[soln, s[[t]][[j]]], soln[[lev]]=s[[t]][[j]]; try[lev+1]; soln[[lev]]=0]]]]; For[lst={1}; n=2, n<=nMax, n++, s=Table[{}, {n}]; For[i=1, i<=n, i++, For[j=1, j<=n, j++, If[i!=j&&PrimeQ[i^2+j^2], AppendTo[s[[i]], j]]]]; soln=Table[0, {n}]; For[cnt=0; i=1, i<=n, i++, soln[[1]]=i; try[2]]; AppendTo[lst, cnt]]; lst
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CROSSREFS
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KEYWORD
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hard,more,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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