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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

T. 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 *)

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

AUTHOR

Thomas Zaslavsky, Mar 14 2010

EXTENSIONS

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

STATUS

approved

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Last modified May 11 05:25 EDT 2021. Contains 343784 sequences. (Running on oeis4.)