

A174257


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


7



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
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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 n3 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 73=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 kfamily 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 ngon.  Frank M Jackson, Oct 30 2022


LINKS



FORMULA

G.f.: x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x1)^2 ).
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((n1)/2) + floor((n1)/3)  floor(n/3).  Mircea Merca, May 14 2013
a(n) = (6*n  13  8*cos(2*n*Pi/3)  3*cos(n*Pi))/12.  Wesley Ivan Hurt, Oct 04 2018


EXAMPLE

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((n1)/2)+floor((n1)/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,


PROG

(PARI) concat(vector(3), Vec(x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x1)^2 ) + O(x^90))) \\ Michel Marcus, Jan 29 2016


CROSSREFS



KEYWORD

nonn,easy,changed


AUTHOR



EXTENSIONS



STATUS

approved



