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) = 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
KEYWORD
nonn,easy
AUTHOR
Thomas Zaslavsky, Mar 14 2010
EXTENSIONS
Information added to name and comments by Thomas Zaslavsky, Apr 24 2010
STATUS
approved