A108576 -id:A108576 - cp's OEIS frontend

A108579 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.

0, 0, 0, 0, 1, 3, 4, 7, 10, 13, 17, 22, 26, 32, 38, 44, 51, 59, 66, 75, 84, 93, 103, 114, 124, 136, 148, 160, 173, 187, 200, 215, 230, 245, 261, 278, 294, 312, 330, 348, 367, 387, 406, 427, 448, 469, 491, 514, 536, 560, 584, 608, 633, 659, 684, 711, 738, 765, 793, 822, 850

Offset: 1

Thomas Zaslavsky, Jun 11 2005

A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6.
It appears that A108579(n) is the number of ordered triples (w,x,y) with components all in {1,...,n} and w+n=2x+3y, as in the Mathematica section.  For related sequences, see A211422. - Clark Kimberling, Apr 15 2012

			a(5) = 1 because there is a unique 3 X 3 magic square, up to symmetry, using the first 9 positive integers.

T. Zaslavsky, Table of n, a(n) for n = 1..10000
Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow: Maple Worksheets and Supporting Documentation.
Yu. V. Chebrakov, Section 2.6.3 in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A108576, A108577, A108578.

Nonzero entries are the second differences of A055328.

(* This program generates a sequence described in the Comments section *)
t[n_] := t[n] = Flatten[Table[-w^2 + x*y + n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 1, 80}]  (* A211506 *)
(* Clark Kimberling, Apr 15 2012 *)

a(n) = floor((1/4)*(n-2)^2)-floor((1/3)*(n-1)). - Mircea Merca, Oct 08 2013

G.f.: x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^3)).

Edited by N. J. A. Sloane, Oct 04 2010

A108577 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having all entries < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 12, 16, 23, 30, 40, 50, 63, 76, 93, 110, 132, 154, 180, 206, 238, 270, 308, 346, 390, 434, 485, 536, 595, 654, 720, 786, 861, 936, 1020, 1104, 1197, 1290, 1393, 1496, 1610, 1724, 1848, 1972, 2108, 2244, 2392, 2540, 2700, 2860

Offset: 1

Thomas Zaslavsky, Jun 11 2005

A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal. The symmetries are those of the square. - Thomas Zaslavsky, Mar 12 2010

			a(10) = 1 because there is only one symmetry type of 3 X 3 magic square with entries 1,...,9.

Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
Matthias Beck and Thomas Zaslavsky, Auxiliary files for "Six little squares".
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).

Cf. A108576, A108578, A108579.

LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5}, 58] (* Mike Sheppard, Feb 04 2025 *)

G.f.: (x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).

a(n) is given by a quasipolynomial of period 12.

A108578 Number of 3 X 3 magic squares with magic sum 3n.

Original entry on oeis.org

0, 0, 0, 0, 8, 24, 32, 56, 80, 104, 136, 176, 208, 256, 304, 352, 408, 472, 528, 600, 672, 744, 824, 912, 992, 1088, 1184, 1280, 1384, 1496, 1600, 1720, 1840, 1960, 2088, 2224, 2352, 2496, 2640, 2784, 2936, 3096, 3248, 3416, 3584, 3752, 3928, 4112, 4288

Offset: 1

Thomas Zaslavsky and Ralf Stephan, Jun 11 2005

nonn

Contribution from Thomas Zaslavsky, Mar 12 2010: (Start)
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6. (End)

			a(5) = 8 because there are 8 3 X 3 magic squares with entries 1,...,9 and magic sum 15.

T. Zaslavsky, Table of n, a(n) for n = 1..10000.
M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - _Thomas Zaslavsky_, Jan 29 2010
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1, -1,1).

Equals 8 times the second differences of A055328.

Cf. A108576, A108577, A108579.

I:=[0,0,0,0,8,24]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Sep 01 2018

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 8, 24}, 50] (* Jean-François Alcover, Sep 01 2018 *)
CoefficientList[Series[8 x^4 (1 + 2 x) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 01 2018 *)

a(n)=(1/9)*(2*n^2-32*n+[144,78,120,126,96,102][(n%18)/3+1])

x='x+O('x^99); concat(vector(4), Vec(8*x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^3)))) \\ Altug Alkan, Sep 01 2018

G.f.: [8*x^5*(1+2*x)] / [(1-x)*(1-x^2)*(1-x^3)].

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - Vincenzo Librandi, Sep 01 2018

Edited by N. J. A. Sloane, Feb 05 2010

Corrected g.f. to account for previous change in parameter n from magic sum s to s/3; by Thomas Zaslavsky, Mar 12 2010

A174257 Number of symmetry classes of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 3, 3, 4, 5, 4, 6, 6, 6, 7, 8, 7, 9, 9, 9, 10, 11, 10, 12, 12, 12, 13, 14, 13, 15, 15, 15, 16, 17, 16, 18, 18, 18, 19, 20, 19, 21, 21, 21, 22, 23, 22, 24, 24, 24, 25, 26, 25, 27, 27, 27, 28, 29, 28, 30, 30, 30, 31, 32, 31, 33, 33, 33, 34, 35, 34, 36, 36, 36, 37

Offset: 1

Thomas Zaslavsky, Mar 14 2010

In a reduced magic square the row, column, and two diagonal sums must all be equal (the "magic sum") and the minimum entry is 0. The maximum entry is necessarily even and = (2/3)*(magic sum). The symmetries are those of the square.
a(n) is a quasipolynomial with period 6.
The second differences of A108577 are a(n/2) for even n and 0 for odd n. The first differences of A108579 are a(n/3).
For n>=3 equals a(n) the number of partitions of n-3 using parts 1 and 2 only, with distinct multiplicities. Example: a(7) = 3 because [2,2], [2,1,1], [1,1,1,1] are such partitions of 7-3=4. - T. Amdeberhan, May 13 2012
a(n) is equal to the number of partitions of n of length 3 with exactly two equal entries (see below example). - John M. Campbell, Jan 29 2016
a(k) + 2 =:t(k), k >= 1, based on sequence A300069, is used to obtain for 2^t(k)*O_{-k} integer coordinates in the quadratic number field Q(sqrt(3)), where O_{-k} is the center of a k-family of regular hexagons H_{-k} forming part of a discrete spiral. See the linked W. Lang paper, Lemma 5, and Table 7. - Wolfdieter Lang, Mar 30 2018
a(n) is equal to the number of incongruent isosceles triangles (excluding equilateral triangles) formed from the vertices of a regular n-gon. - Frank M Jackson, Oct 30 2022

			From _John M. Campbell_, Jan 29 2016: (Start)
For example, there are a(16)=7 partitions of 16 of length 3 with exactly two equal entries:
  (14,1,1) |- 16
  (12,2,2) |- 16
  (10,3,3) |- 16
   (8,4,4) |- 16
   (7,7,2) |- 16
   (6,6,4) |- 16
   (6,5,5) |- 16
(End)

Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, Vol. 13 (2010), Article 10.6.2.
Wolfdieter Lang, On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A108576, A108577, A174256, A300069.

seq(floor((n-1)/2)+floor((n-1)/3)-floor(n/3),n=1..100) # Mircea Merca, May 14 2013

Rest@ CoefficientList[Series[x^4 (1 + 2 x)/((1 + x) (1 + x + x^2) (x - 1)^2), {x, 0, 76}], x] (* Michael De Vlieger, Jan 29 2016 *)
Table[Length@Select[Length/@Union/@IntegerPartitions[n, {3}], # == 2 &], {n,
  1, 100}] (* Frank M Jackson, Oct 30 2022 *)

concat(vector(3), Vec(x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x-1)^2 ) + O(x^90))) \\ Michel Marcus, Jan 29 2016

G.f.: x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x-1)^2 ).
a(n) = (1/8)*A174256(n).
a(n) = floor((n-1)/2) + floor((n-1)/3) - floor(n/3). - Mircea Merca, May 14 2013
a(n) = A300069(n-1) + 3*floor((n-1)/6), n >= 1. Proof via g.f.. - Wolfdieter Lang, Feb 24 2018
a(n) = (6*n - 13 - 8*cos(2*n*Pi/3) - 3*cos(n*Pi))/12. - Wesley Ivan Hurt, Oct 04 2018

Information added to name and comments by Thomas Zaslavsky, Apr 24 2010

A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").

Original entry on oeis.org

72, 288, 936, 2592, 5760, 11520, 20952, 35712, 57168, 88272, 131112, 189504, 265752, 365760, 492480, 653040, 851472, 1096416, 1392768, 1751904, 2178864, 2687184, 3283632, 3983760, 4794984, 5736528, 6816456, 8056224, 9466128

Offset: 10

Thomas Zaslavsky, Feb 21 2010

nonn

a(n) is given by a quasipolynomial of degree 5 and period 60.

Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.

Thomas Zaslavsky, Table of n, a(n) for n = 10..10000.
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1).

A173547 counts the same squares by magic sum.

Cf. A108576, A108577, A108578, A108579, A173548, A173549.

G.f.: 72 * x^2/(1-x)^2 * { x^5/[(1-x)^3*(1-x^2)] - 2x^5/[(1-x)*(1-x^2)^2] - x^5/[(1-x)^2*(1-x^3)] - 2x^6/[(1-x)*(1-x^2)*(1-x^3)] - x^6/(1-x^2)^3 - x^7/[(1-x^2)^2*(1-x^3)] + x^5/[(1-x)*(1-x^4)] + 2x^5/[(1-x^2)*(1-x^3)] + 2x^6/[(1-x^2)*(1-x^4)] + x^6/(1-x^3)^2 + x^7/[(1-x^2)*(1-x^5)] + x^7/[(1-x^3)*(1-x^4)] + x^8/[(1-x^3)*(1-x^5)] - x^5/(1-x^5) }. - Thomas Zaslavsky, Mar 03 2010

A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.

Original entry on oeis.org

72, 144, 288, 576, 864, 1440, 2088, 3024, 3888, 5904, 6984, 9432, 12168, 14904, 17928, 23832, 26784, 33048, 39672, 46584, 53640, 65592, 72504, 85248, 98928, 111816, 125208, 147528, 160632, 182808, 206424, 229176, 252648, 287928, 310752

Offset: 15

Thomas Zaslavsky, Feb 21 2010, Feb 24 2010, Mar 03 2010

nonn

In a semimagic squares the row and column sums must all be equal to the magic sum. a(n) is given by a quasipolynomial of degree 4 and period 840.

a(15) is the first term because the values 1,...,9 make magic sum 15. [From Thomas Zaslavsky, Mar 03 2010]

Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.

Thomas Zaslavsky, Table of n, a(n) for n=15..10000.
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
Index entries for linear recurrences with constant coefficients, signature (-2, -3, -2, 0, 3, 6, 8, 9, 7, 3, -4, -10, -15, -16, -14, -8, 0, 8, 14, 16, 15, 10, 4, -3, -7, -9, -8, -6, -3, 0, 2, 3, 2, 1).

A173546 counts the same squares by upper bound on the entries. Cf. A108576, A108577, A108578, A108579, A173548, A173549.

G.f.: 72 * x^3/(1-x)^3 * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }

A174256 Number of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

Original entry on oeis.org

0, 0, 0, 8, 16, 8, 24, 24, 24, 32, 40, 32, 48, 48, 48, 56, 64, 56, 72, 72, 72, 80, 88, 80, 96, 96, 96, 104, 112, 104, 120, 120, 120, 128, 136, 128, 144, 144, 144, 152, 160, 152, 168, 168, 168, 176, 184, 176, 192, 192, 192, 200, 208, 200, 216, 216, 216, 224, 232, 224

Offset: 1

Thomas Zaslavsky, Mar 14 2010

In a reduced magic square the row, column, and two diagonal sums must all be equal (the "magic sum") and the minimum entry is 0. The maximum entry is necessarily even and = (2/3)*(magic sum).
a(n) is a quasipolynomial with period 6.
The second differences of A108576 are a(n/2) for even n and 0 for odd n. The first differences of A108578 are a(n).

Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A108576, A108577, A174257.

Take[CoefficientList[Series[(8x^8 (2x^2+1))/((x^4-1)(x^6-1)),{x,0,120}],x],{1,-1,2}] (* Harvey P. Dale, Aug 07 2017 *)

a(n) = 8*A174257(n).
G.f.: 8*x^4 * (2*x+1) / ((x^2-1) * (x^3-1)). [amended by Georg Fischer, Apr 17 2020]
a(n) = 2*(6*n - 13 - 8*cos(2*n*Pi/3) - 3*cos(n*Pi))/3. - Wesley Ivan Hurt, Oct 04 2018

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A108579 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.

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A108577 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having all entries < n.

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A108578 Number of 3 X 3 magic squares with magic sum 3n.

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A174257 Number of symmetry classes of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

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A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").

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A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.

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A174256 Number of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

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