A033812 -id:A033812 - cp's OEIS frontend

A126652 A 3 X 3 magic square with magic sum 75: the Loh-Shu square A033812 multiplied by 5.

40, 5, 30, 15, 25, 35, 20, 45, 10

Offset: 1

Parthasarathy Nambi, Feb 08 2007

The magic sum is 75. See Figure C.6 in Appendix C in Maya Ahmed's PhD thesis.

			The magic square is:
  40  5 30
  15 25 35
  20 45 10

Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004. PhD thesis.
Maya Mohsin Ahmed, How many squares are there, Mr. Franklin?: Constructing and Enumerating Franklin Squares, Amer. Math. Monthly, Vol. 111, 2004, pp. 394-410.
Index entries for sequences related to magic squares

Cf. A126653 = 3*A033812.

a(n) = 5*A033812(n). - M. F. Hasler, Jan 26 2019

Edited by M. F. Hasler, Jan 26 2019

A126653 A 3 X 3 magic square with magic sum 45: the Loh-Shu square A033812 multiplied by 3.

Original entry on oeis.org

24, 3, 18, 9, 15, 21, 12, 27, 6

Offset: 1

Parthasarathy Nambi, Feb 08 2007

The magic sum is 45. See Figure C.6 in Appendix C in Maya Ahmed's PhD thesis.

			The magic square is:
24  3 18
9  15 21
12 27  6

Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004. PhD thesis.
Maya Mohsin Ahmed, How many squares are there, Mr. Franklin?: Constructing and Enumerating Franklin Squares, Amer. Math. Monthly, Vol. 111, 2004, pp. 394-410.
Index entries for sequences related to magic squares

Cf. A126652 for 5*A033812.

a(n) = 3*A033812(n). - M. F. Hasler, Jan 26 2019

Edited by M. F. Hasler, Jan 26 2019

A126648 A 3 X 3 magic square with magic sum 123 and entries congruent to 1 (mod 10): equals 10*A033812 - 9.

Original entry on oeis.org

71, 1, 51, 21, 41, 61, 31, 81, 11

Offset: 1

Parthasarathy Nambi, Feb 08 2007

The magic sum is 123. See Figure C.4 in Appendix C in Maya Ahmed's PhD thesis.

			The magic square is:
  71  1 51
  21 41 61
  31 81 11

Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004. PhD thesis.
Maya Mohsin Ahmed, How many squares are there, Mr. Franklin?: Constructing and Enumerating Franklin Squares, Amer. Math. Monthly, Vol. 111, 2004, pp. 394-410
Index entries for sequences related to magic squares

Cf. A033812.

a(n) = 10*A033812(n) - 9. - M. F. Hasler, Jan 26 2019

Edited by M. F. Hasler, Jan 26 2019

A126709 The Loh-Shu 3 x 3 magic square, variant 2.

Original entry on oeis.org

4, 9, 2, 3, 5, 7, 8, 1, 6

Offset: 1

Parthasarathy Nambi, Feb 12 2007

The magic sum is 15. See Figure 1.2 on page 10 in Maya Ahmed's PhD thesis. This magic square is referred to Loh-Shu magic square and is attributed to the legendary Fu Xi (Fuh-Hi).

Sequence A033812 lists the same magic square with rows in reversed order, i.e., vertically flipped. - M. F. Hasler, Jan 26 2019

			The magic square is:
  4 9 2
  3 5 7
  8 1 6

Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004.
Alexander Poddiakov, Self-Similar Structures of Nontransitive Dice Sets: Examples of Nested "Rock-Paper-Scissors" Relations Based on Numbers from The Lo Shu Magic Square,
Index entries for sequences related to magic squares

A033812 is the same magic square with rows listed in reverse order.
Other magic 3 X 3 squares: A126648, A126652, A126653, A126654.
Larger magic squares: A126647, A126649, A126651, A126650.

Edited by M. F. Hasler, Jan 26 2019

A091281 Central term in powers of the Lo-Shu Magic Square as a matrix.

Original entry on oeis.org

1, 5, 91, 1125, 17259, 253125, 3806091, 56953125, 854518059, 12814453125, 192222105291, 2883251953125, 43248906698859, 648731689453125, 9730978399444491, 145964630126953125, 2189469525287839659, 32842041778564453125, 492630628439671823691, 7389459400177001953125

Offset: 0

Gary W. Adamson, Dec 28 2003

a(n)/a(n-1) tends to 15, the "Magic Number" of the Lo-Shu Magic Square.

There are a total of 8 variations of the Lo-Shu magic square by rotations and/or reflections. Four of the variations (those with 4, 5, 6 or 6, 5, 4 in the diagonal), have a(2) = 91. The other 4 variations (those with 2, 5, 8 or 8, 5, 2 in the diagonal - lower left to upper right) have a(2) = 59, but otherwise, a(n) for the latter sequence (central term in analogous powers of those matrices) = A091281(n).

			a(2) = 91 since M^2 = [91, 67, 67 / 67, 91, 67 / 67, 67, 91].

Index entries for linear recurrences with constant coefficients, signature (15,24,-360).

Cf. A033812.

a(n)=([8,1,6;3,5,7;4,9,2]^n)[2,2] \\ Charles R Greathouse IV, Dec 14 2011

The Lo-Shu magic square square as a 3 X 3 matrix is: [8, 1, 6 / 3, 5, 7 / 4, 9, 2] = M. Then a(n) = central term in M^n.
a(2*k+1) = 5*15^(2*k). E.g. a(5) = 253125 = 5*15^4.
a(n) = (1/69)*(23*15^n - 2*24^floor((n+1)/2) + 2*24^floor((n+2)/2)). - Ralf Stephan, Dec 02 2004
G.f.: -(8*x^2+10*x-1) / ((15*x-1)*(24*x^2-1)). - Colin Barker, Dec 10 2012

a(12)-a(19) from Charles R Greathouse IV, Dec 14 2011

cp's OEIS Frontend

A126652 A 3 X 3 magic square with magic sum 75: the Loh-Shu square A033812 multiplied by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A126653 A 3 X 3 magic square with magic sum 45: the Loh-Shu square A033812 multiplied by 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A126648 A 3 X 3 magic square with magic sum 123 and entries congruent to 1 (mod 10): equals 10*A033812 - 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A126709 The Loh-Shu 3 x 3 magic square, variant 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A091281 Central term in powers of the Lo-Shu Magic Square as a matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions