A072129 - OEIS

%I #13 Sep 03 2024 16:12:30

%S 1,0,0,0,6,0,96,272,1408,61622,33736,356606,86520774,192570133,

%T 1560696233

%N Number of distinct ways of arranging the squares {1,4,9,...,(2n)^2} in a circle so that the sum of each two adjacent entries is a prime.

%e a(5)=6 because there are 6 essentially different ways: {1, 4, 9, 64, 49, 100, 81, 16, 25, 36}, {1, 4, 49, 64, 9, 100, 81, 16, 25, 36}, {1, 16, 81, 100, 9, 4, 49, 64, 25, 36}, {1, 16, 81, 100, 9, 64, 49, 4, 25, 36}, {1, 16, 81, 100, 49, 4, 9, 64, 25, 36} and {1, 16, 81, 100, 49, 64, 9, 4, 25, 36}

%t $RecursionLimit=500; try[lev_] := Module[{t, j}, If[lev>2n, (*then make sure the sum of the first and last is prime*) If[PrimeQ[soln[[1]]^2+soln[[2n]]^2]&&soln[[2]]<=soln[[2n]], (*Print[soln]; *) 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={}; n=1, n<=7, n++, s=Table[{}, {2n}]; For[i=1, i<=2n, i++, For[j=1, j<=2n, j++, If[i!=j&&PrimeQ[i^2+j^2], AppendTo[s[[i]], j]]]]; soln=Table[0, {2n}]; soln[[1]]=1; cnt=0; try[2]; AppendTo[lst, cnt]]; lst

%Y Cf. A051252, A073451.

%K nonn,more

%O 1,5

%A _Santi Spadaro_, Jun 25 2002

%E Corrected and extended by _T. D. Noe_, Jul 03 2002

%E a(13) corrected and a(14)-a(15) from _Sean A. Irvine_, Sep 03 2024