A052457 -id:A052457 - cp's OEIS frontend

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

Eric W. Weisstein

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

			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.

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 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

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

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

$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 *)

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 2A068606 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

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

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

A052458 Number of trimagic series for squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 121, 126, 0, 31187, 2226896, 17265701, 0, 69303997733

Offset: 1

Eric W. Weisstein

Asymptotic results are presented in Quist for magic cube and hypercube series, bimagic series, and trimagic series. - Jonathan Vos Post, Jun 04 2013

M. Kraitchik, Mathematical Recreations, 1942, see Section 7.10.

Christian Boyer, Multimagic Squares
Michael Quist, Asymptotic enumeration of magic series, arXiv:1306.0616v1 [math.CO], June 3, 2013.
Eric Weisstein's World of Mathematics, Multimagic Series

Cf. A052456, A052457, A090653, A092312, A090037.

Corrected and extended Nov 15 2003, using the values of a(3) through a(12) from Christian Boyer's web site. - N. J. A. Sloane
One more term from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
One further term from Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005
a(15) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A090037 Number of tetramagic (4-multimagic) series for squares of order n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106, 555, 0, 0, 235275

Offset: 1

Eric W. Weisstein, Nov 19 2003

Christian Boyer, Multimagic Squares
Eric Weisstein's World of Mathematics, Magic Series

Cf. A052456, A052457, A052458, A090653, A092312, A106646.

One more term from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
One further term from Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005
a(15)=0 proved by Robert Gerbitz and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007
a(16) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A090653 Number of bimagic series for cubes of order n.

Original entry on oeis.org

1, 0, 4, 8, 272, 25270, 5152529, 1594825624, 651151145259, 347171191981324

Offset: 1

Christian Boyer, Feb 05 2004

Christian Boyer, Multimagic Cubes

Cf. A052457, A092312, A090037.

Cf. A052457, A090037, A092312.

a(9) and a(10) computed by Walter Trump and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007

a(10) corrected by Walter Trump, communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A092312 Number of trimagic series for cubes of order n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 161, 17218, 363949, 0

Offset: 1

Christian Boyer, Feb 14 2004

Christian Boyer, Multimagic Cubes

Cf. A052457, A052458, A090653, A090037.

a(9) computed by Gildas Guillemot; a(10)=0 because a(4k+2)=0. - Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007

A106646 Number of pentamagic (5-multimagic) series for squares of order n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 13

Offset: 1

Christian Boyer (cboyer(AT)club-internet.fr), May 30 2005

Christian Boyer, Multimagic Squares
Eric Weisstein's World of Mathematics, Magic Series

Cf. A090037.

Cf. A052456, A052457, A052458, A090037.

a(15)=0 proved by Robert Gerbitz and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007

a(16) computed by Michael Quist, and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Feb 06 2009

A381001 Georges Pfeffermann's 1890 bimagic square of order 8, read by rows.

Original entry on oeis.org

56, 34, 8, 57, 18, 47, 9, 31, 33, 20, 54, 48, 7, 29, 59, 10, 26, 43, 13, 23, 64, 38, 4, 49, 19, 5, 35, 30, 53, 12, 46, 60, 15, 25, 63, 2, 41, 24, 50, 40, 6, 55, 17, 11, 36, 58, 32, 45, 61, 16, 42, 52, 27, 1, 39, 22, 44, 62, 28, 37, 14, 51, 21, 3

Offset: 1

Paolo Xausa, Feb 13 2025

This is the first known bimagic square. It contains all numbers from 1 to 64; the magic sum is 260 and, when each number is squared, the magic sum is 11180.

			The magic square is:
  [56 34  8 57 18 47  9 31]
  [33 20 54 48  7 29 59 10]
  [26 43 13 23 64 38  4 49]
  [19  5 35 30 53 12 46 60]
  [15 25 63  2 41 24 50 40]
  [ 6 55 17 11 36 58 32 45]
  [61 16 42 52 27  1 39 22]
  [44 62 28 37 14 51 21  3]

Christian Boyer, Bimagic squares.
Brady Haran and Matt Parker, A Magic Square Breakthrough, YouTube Numberphile video, 2025.
Index entries for sequences related to magic squares

Cf. A052457, A111155, A380966, A381002.

cp's OEIS Frontend

A052456 Number of magic series of order n.

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A052458 Number of trimagic series for squares of order n.

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A090037 Number of tetramagic (4-multimagic) series for squares of order n.

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A090653 Number of bimagic series for cubes of order n.

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A092312 Number of trimagic series for cubes of order n.

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A106646 Number of pentamagic (5-multimagic) series for squares of order n.

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A381001 Georges Pfeffermann's 1890 bimagic square of order 8, read by rows.

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