Search: a300069 -id:a300069
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A174257
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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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+10
7
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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
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x-1)^2 ).
a(n) = floor((n-1)/2) + floor((n-1)/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
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EXAMPLE
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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)
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MAPLE
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seq(floor((n-1)/2)+floor((n-1)/3)-floor(n/3), n=1..100) # Mircea Merca, May 14 2013
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MATHEMATICA
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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,
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A300075
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Period 6: repeat [0, 1, 1, 2, 2, 2].
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+10
2
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0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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a(n) = floor(((n+1) (mod 6))/2) + 2*floor((n (mod 6))/5), n >= 0.
G.f.: x*(1 + x + 2*x^2*(1 + x + x^2))/(1 - x^6).
a(n) = (4 - 2*cos(n*Pi/3) - cos(2*n*Pi/3) - cos(n*Pi) - sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Oct 04 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A300290
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Period 6: repeat [0, 1, 2, 2, 3, 3].
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+10
1
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0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = n (mod 6) - floor((n (mod 6))/3) - floor((n (mod 6))/5), n >= 0.
G.f.: x*(1 + x*(2 + 3*x^2)*(1 + x))/(1 - x^6).
a(n) = (11 - 5*cos(n*Pi/3) - 5*cos(2*n*Pi/3) - cos(n*Pi) - 3*sqrt(3)*sin(n*Pi/3) - sqrt(3)*sin(2*n*Pi/3))/6. - Wesley Ivan Hurt, Oct 04 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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