A052456 Number of magic series of order n.

1, 1, 2, 8, 86, 1394, 32134, 957332, 35154340, 1537408202, 78132541528, 4528684996756, 295011186006282, 21345627856836734, 1698954263159544138, 147553846727480002824, 13888244935445960871352, 1408407905312396429259944, 153105374581396386625831530

Offset

0,3

Comments

Henry Bottomley's narrowing gap could be confirmed for 2 < n <= 64. - Walter Trump, 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, May 05 2006

References

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

Links

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..150 (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, Magic Series

Eric Weisstein's World of Mathematics, Multimagic Series

Robert Gerbicz, Walter Trump, First 150 terms

Robert Gerbicz, C-program to generate the sequence

Formula

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. - Henry Bottomley, Feb 25 2002.

It seems a(n) (at least for 2<n<=36) 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, Feb 25 2002.

a(n) ~ sqrt(3) * exp(n-1/2) * n^(n-3) / Pi. - Vaclav Kotesovec, Sep 05 2014

Example

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.

Mathematica

$RecursionLimit = 1000; b[n_, i_, t_] /; i < t || n < t*((t + 1)/2) || n > t*((2*i - t + 1)/2) = 0; b[0, _, _] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i - 1, t] + If[n < i, 0, b[n - i, i - 1, t - 1]]; a[_, 0] = 1; a[0, _] = 0; a[n_, k_] :=  With[{s = k*(k*n + 1)}, If[Mod[s, 2] == 1, 0, b[s/2, k*n, k]]]; a[n_] := a[n] = a[n, n]; Table[Print[a[n]]; a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 15 2013, after Alois P. Heinz *)

Crossrefs

Cf. A007785, A052457, A052458. A100568 is the same sequence times n!.

Main diagonal of A204459. - Alois P. Heinz, Jan 18 2012

Sequence in context: A136647 A306001 A261730 * A276991 A000532 A333366

Adjacent sequences: A052453 A052454 A052455 * A052457 A052458 A052459

Keyword

nonn,nice

Author

Eric W. Weisstein

Extensions

Edited and ten more terms from Henry Bottomley, Feb 16 2002

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

Status

approved