A052458 -id:A052458 - cp's OEIS frontend

A087637 Erroneous version of A052458.

1, 0, 0, 2, 2, 0, 0, 115, 41, 0, 961

Offset: 1

dead

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

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

A052457 Number of bimagic series for squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 8, 98, 1844, 38039, 949738, 24643236, 947689757, 45828982764, 2151748695931, 123821075526032, 8131094055190149, 573957471153552576, 44010987379157415768, 3655486139293429450720, 333633403912637510806972

Offset: 1

Eric W. Weisstein

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

Cf. A052456, A052458, A090037, A106646, A090653.

Extended Nov 08 2003, using the values of a(3) through a(11) from Christian Boyer's web site. - N. J. A. Sloane.
a(12) from Christian Boyer (cboyer(AT)club-internet.fr), Nov 05 2004
a(13)-a(15) computed by Lorenz Schlangen and later independently confirmed by Walter Trump. a(16) computed by Walter Trump. - Christian Boyer (cboyer(AT)club-internet.fr), Oct 04 2005
a(17) computed by Walter Trump and communicated by Christian Boyer (cboyer(AT)club-internet.fr), Apr 02 2007
a(18)-a(19) computed by Michael Quist 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

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

A381002 Gaston Tarry's 1905 trimagic square of order 128, read by rows.

Original entry on oeis.org

16132, 130, 16381, 127, 16128, 382, 15873, 387, 13632, 2750, 13761, 2627, 13508, 2882, 13373, 3007, 8452, 7810, 8701, 7807, 8448, 8062, 8193, 8067, 11072, 5310, 11201, 5187, 10948, 5442, 10813, 5567, 10028, 6314, 10197, 6231, 9944, 6486, 9769, 6571, 13080, 3222, 13289

Offset: 1

Paolo Xausa, Feb 13 2025

This is the first known trimagic square. It contains all numbers from 1 to 16384. The magic sum is 1048640; when each number is squared, the magic sum is 11454294720; and when each number is cubed, the magic sum is 140754668748800.

Terms are taken from Christian Boyer's Multimagie website (see links).

			The magic square is:
  [16132   130 16381   127 16128 ... 11854  4301 12111  4148 12210]
  [  128 16382   129 16131   388 ...  4402 12209  4147 12112  4302]
  [16002   260 15999   509 16254 ... 12240  4431 11981  4530 11828]
  [  510 16000   259 16001     2 ...  4276 11827  4529 11982  4432]
  [  257 16003   512 15998   253 ...  4175 11984  4430 11825  4531]
     ...   ...   ...   ...   ... ...   ...   ...   ...   ...   ...
  [ 4642 11684  4831 11613  5086 ...  7496  9159  7237  9018  7356]
  [ 4829 11615  4644 11682  4897 ...  7611  9020  7354  9157  7239]
  [11681  4643 11616  4830 11357 ...  8903  7240  9158  7353  9019]
  [ 4959 11485  5026 11300  4771 ...  7225  8890  7484  8775  7621]
  [11299  5025 11486  4960 11743 ...  9029  7622  8776  7483  8889]

Paolo Xausa, Table of n, a(n) for n = 1..16384
Christian Boyer, Trimagic square, 128th-order.
Brady Haran and Matt Parker, A Magic Square Breakthrough, YouTube Numberphile video, 2025.
Index entries for sequences related to magic squares

Cf. A052458, A380966, A381001.

cp's OEIS Frontend

A087637 Erroneous version of A052458.

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A052456 Number of magic series of order n.

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A052457 Number of bimagic 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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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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A381002 Gaston Tarry's 1905 trimagic square of order 128, read by rows.

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