0,3

Henry Bottomley's narrowing gap could be confirmed for 2 < n <= 64 - Walter Trump (w(AT)trump.de), Jan 21 2005

A new algorithm was found by Robert Gerbicz. Now the enumeration of magic series of orders greater than 100 is possible. - Walter Trump (w(AT)trump.de), May 05 2006

M. Kraitchik, Magic Series. Section 7.13.3 in Mathematical Recreations, New York, W. W. Norton, pp. 143 and 183-186, 1942.

T. D. Noe, Table of n, a(n) for n=0..100 (from Gerbicz and Trump)

H. Bottomley, Partition and composition calculator

H. Bottomley and W. Trump, First 36 terms

Walter Trump, Magic Squares.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Robert Gerbicz, Walter Trump, First 150 terms

Robert Gerbicz, C-program to generate the sequence

a(n) = A067059(n, n*(n-1)) = r(n, n*(n-1), n^2*(n-1)/2) where r(n, m, k) is a restricted partition function giving the number of partitions of k into at most n positive parts each no more than m; (at least for 2<n< = 36) it seems a(n) is in the narrowing gap between C(n^2, n)*1.381976597885.../n^(5/2) and C(n^2, n)*sqrt(6/(pi*n^2*(n-1)*(n^2+1))): cf. A068606 and assuming the peak of a normal distribution = 1/sqrt(variance*2*pi) - Henry Bottomley (se16.btinternet.com), Feb 25 2002.

a(3)=8 since a magic square of order 3 would require a row sum of 15=(1+2+...+9)/3 and there are 8 ways of writing 15 as the sum of three distinct positive numbers up to 9: 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6.

Cf. A052457, A052458. A100568 is the same sequence times n!.

Sequence in context: A120820 A134089 A136647 this_sequence A000532 A083831 A134245

Adjacent sequences: A052453 A052454 A052455 this_sequence A052457 A052458 A052459

nonn,nice

Eric Weisstein (eric(AT)weisstein.com)

Edited and ten more terms from Henry Bottomley (se16(AT)btinternet.com), Feb 16 2002

Terms through a(36) added to attached web page, Feb 04 2005