A064817 Maximal number of squares among the n-1 numbers p_i + p_{i+1}, 1 <= i <= n-1, where (p_1, ..., p_n) is any permutation of (1, ..., n).

0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 16, 17, 18, 19, 20, 22, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Author

N. J. A. Sloane, Oct 23 2001

Keywords

• nonn
• nice

Comments

a(n) < n by definition, but if we counted the sum p_n + p_1, we could get a(n) = n for 32 <= n <= 49 (see A071984). - David Wasserman, Aug 20 2002

Can be formulated as a traveling salesman problem on a complete graph with node set {0, 1, ..., n} and edge cost -1 if i + j is a square, 0 otherwise. - Rob Pratt, Nov 07 2012

a(n) = n - 1 for 25 <= n <= 500, computed by solving corresponding TSP. - Rob Pratt, Nov 07 2012

Examples

			n=8: take 2,7,8,1,3,6,4,5 to get 5 squares: 2+7, 8+1, 1+3, 3+6, 4+5; a(8) = 5.
(1,8,9,7,2,14,11,5,4,12,13,3,6,10) gives 12 squares and no permutation of (1..14) gives more, so a(14)=12.
		

References

• Bernardo Recamán Santos, Challenging Brainteasers, Sterling, NY, 2000, page 71, shows a(15) = 14 using 9,7,2,14,11,5,4,12,13,3,6,10,15,1,8.

Crossrefs

Cf. A035106, A064764, A064796, A064797, A071983, A071984.

Programs

Mathematica

a[n_] := Which[n == 1, 0, n > 30, n - 1, True, tour = FindShortestTour[Range[n], DistanceFunction -> Function[{i, j}, If[IntegerQ[Sqrt[i + j]], -1, 0]]] // Last; cnt = 0; Do[If[IntegerQ[Sqrt[tour[[i]] + tour[[i + 1]]]], cnt++], {i, 1, n}]; cnt]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 69}] (* Jean-François Alcover, Nov 04 2016 *)

Extensions

More terms from Vladeta Jovovic, Oct 23 2001

More terms from John W. Layman and Charles K. Layman (cklayman(AT)juno.com), Nov 07 2001

More terms from David Wasserman, Aug 20 2002

More terms from Rob Pratt, Nov 07 2012