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

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

1,5

Comments

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

Links

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).

Formula

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

a(n) = (1/8)*A174256(n).

a(n) = (1/90)*((n mod 6) + ((n+1) mod 6) - 29*((n+2) mod 6) - 29*((n+3) mod 6) - 14*((n+4) mod 6) - 14*((n+5) mod 6)) + (1/30)*Sum_{k=0..n} (-14*(k mod 6) + ((k+1) mod 6) + ((k+2) mod 6) + ((k+3) mod 6) + ((k+4) mod 6) + 16*((k+5) mod 6)), with n >= 0. - Paolo P. Lava, Mar 22 2010

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

Example

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)

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A108576, A108577, A174256, A300069.

Sequence in context: A299966 A302395 A110425 * A105637 A029161 A035384

Adjacent sequences: A174254 A174255 A174256 * A174258 A174259 A174260

Keyword

nonn,easy,changed

Author

Thomas Zaslavsky, Mar 14 2010

Extensions

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

Status

approved