William Walkington - cp's OEIS frontend

A271104 Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2.

1, 0, 1, 4293, 23161722048, 2627518340149999905600

Offset: 1

William Walkington, Mar 30 2016

Initially based on empirical observations by the author, the results for the magic tori of orders 1 to 4, have since been computed and confirmed by Walter Trump. The results for the magic tori of order 5, and for the semi-magic tori of orders 4 and 5, have been computed by Walter Trump. The result for the order 6 is deduced from Artem Ripatti's findings (cf. A271103).

A semi-magic torus differs from a magic torus in that there are no magic intersections of magic diagonals, and in consequence only semi-magic squares are displayed on its surface.

Dwane Campbell, Analysis of order-4 magic squares, (2013).
Dwane Campbell, Order-4 squares grouped by base square quartets, (2013).
Dwane Campbell, Features in order-4 magic squares, (2013).
William Walkington, 255 tores magiques d'ordre 4, et 1 tore magique d'ordre 3, (2011).
William Walkington, Passage du carré au tore magique, (2011).
William Walkington, 255 fourth-order magic tori, and 1 third-order magic torus, (2012).
William Walkington, From the magic square to the magic torus, (2012).
William Walkington, (using findings computed by _Walter Trump_), 251 449 712 fifth-order magic tori, (2012).
William Walkington, A new census of fourth-order magic squares, (2012).
William Walkington, Table of fourth-order magic tori, (2012).

Cf. A006052, A270876, A271103.

a(n) = A271103(n)/ n^2.

a(6) added by William Walkington, Jul 18 2018

A271103 Number of magic and semi-magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

Original entry on oeis.org

1, 0, 9, 68688, 579043051200, 94590660245399996601600

Offset: 1

William Walkington, Mar 30 2016

A semi-magic square differs from a magic square in that at least one of its main diagonals does not sum to the magic constant. [Walter Trump]
The number of order 4 magic and semi-magic squares was computed by Mutsumi Suzuki, and could be found on his former web site. Mutsumi Suzuki's pages are now in the Internet Archive.
The number of order 5 magic and semi-magic squares was computed by Walter Trump in March 2000.
The number of order 6 magic and semi-magic squares was calculated by Artem Ripatti in April 2018, and published in his paper dated July 10, 2018. - William Walkington, Jul 17 2018

Artem Ripatti, On the number of semi-magic squares of order 6, arXiv:1807.02983 [math.CO], 2018. See Table 1 p. 2.
Mutsumi Suzuki, Semi-Magic Squares of 4 x 4, first "captured" by the Internet Archive on the 5th October 1999.
Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Index entries for sequences related to magic squares

Cf. A006052, A270876, A271104.

a(n) = A271104(n)* n^2.

a(6) added by William Walkington, Jul 17 2018

A270824 Period 16: repeat [0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1].

Original entry on oeis.org

0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1

Offset: 0

William Walkington, Mar 23 2016

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1,1).

Cf. A270823.

PadRight[{},120,{0,1,1,0,2,3,3,2,3,2,2,3,1,0,0,1}] (* Harvey P. Dale, Sep 16 2017 *)

a(n)=[0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1][n%16+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = (floor((n+4)/4) - [floor(n/4) + floor((n+4)/4)] * [(floor((n+12)/8)) mod 2 * (floor((n+3)/2)) mod 2 + (floor((n+4)/8)) mod 2 * (floor((n+1)/2)) mod 2]) mod 4.
From Chai Wah Wu, Jun 04 2016: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n > 8.
G.f.: x*(1 - x^2 + 2*x^3 + x^4 - x^6 + x^7)/((1 - x)*(1 + x^8)). (End)

A270823 Period 16: repeat [0,2,3,1,1,3,2,0,1,3,2,0,0,2,3,1].

Original entry on oeis.org

0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 1, 3, 2, 0, 1, 3, 2, 0, 0, 2, 3, 1

Offset: 0

William Walkington, Mar 23 2016

Decimal expansion of 700330007/30300000303. - Elmo R. Oliveira, Feb 21 2024

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,0,-1,1,-1,1).

Cf. A270824.

PadRight[{},120,{0,2,3,1,1,3,2,0,1,3,2,0,0,2,3,1}] (* Harvey P. Dale, May 17 2017 *)

a(n)=[0,2,3,1,1,3,2,0,1,3,2,0,0,2,3,1][n%16+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = (n+1+(15-2n)*[(floor((n+12)/8)) mod 2*(floor((n+3)/2)) mod 2+(floor((n+4)/8)) mod 2*(floor((n+1)/2)) mod 2]) mod 4.
From Chai Wah Wu, Jun 04 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-8) + a(n-9) - a(n-10) + a(n-11) for n > 10.
G.f.: x*(2 + x + x^3 - x^5 + 2*x^8 + x^9)/((1 - x)*(1 + x^2)*(1 + x^8)). (End)

A270740 Period 9: repeat 0,1,2,2,0,1,1,2,0.

Original entry on oeis.org

0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0

Offset: 0

William Walkington, Mar 22 2016

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,1).

Cf. A002264 (floor(n/3)), A010872 (n mod 3).

Table[Mod[n - Floor[n/3], 3], {n, 0, 120}] (* Michael De Vlieger, Mar 25 2016 *)
PadRight[{},120,{0,1,2,2,0,1,1,2,0}] (* Harvey P. Dale, Jan 16 2019 *)

a(n) = (n - n\3) % 3; \\ Michel Marcus, Mar 22 2016

a(n) = (n - floor(n/3)) mod 3.
From Chai Wah Wu, Jun 04 2016: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-7) for n > 6.
G.f.: x*(1 + x - x^3 + 2*x^4)/((1 - x)*(1 + x^3 + x^6)). (End)

A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

Original entry on oeis.org

1, 0, 1, 255, 251449712

Offset: 1

William Walkington, Mar 24 2016

Initially based on empirical observations by William Walkington, the results for the orders 1 to 4 have since been computed and confirmed by Walter Trump. The results for the order 5 have been computed by Walter Trump.

Dwane Campbell, Analysis of order-4 magic squares, (2013).
Dwane Campbell, Order-4 squares grouped by base square quartets, (2013).
Dwane Campbell, Features in order-4 magic squares, (2013).
William Walkington, 255 tores magiques d'ordre 4, et 1 tore magique d'ordre 3, (2011).
William Walkington, Passage du carré au tore magique, (2011).
William Walkington, 255 fourth-order magic tori, and 1 third-order magic torus, (2012).
William Walkington, From the magic square to the magic torus, (2012).
William Walkington, (using findings computed by _Walter Trump_), 251 449 712 fifth-order magic tori, (2012).
William Walkington, A new census of fourth-order magic squares, (2012).
William Walkington, Table of fourth-order magic tori, (2012).
William Walkington, 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4, (2023).

Cf. A006052, A271103, A271104.

cp's OEIS Frontend

User: William Walkington

A271104 Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2.

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A271103 Number of magic and semi-magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

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A270824 Period 16: repeat [0, 1, 1, 0, 2, 3, 3, 2, 3, 2, 2, 3, 1, 0, 0, 1].

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A270823 Period 16: repeat [0,2,3,1,1,3,2,0,1,3,2,0,0,2,3,1].

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A270740 Period 9: repeat 0,1,2,2,0,1,1,2,0.

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A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

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