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Search: a300069 -id:a300069
Displaying 1-3 of 3 results found. page 1
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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. +10
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 (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
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
Cf. A108576, A108577, A174256, A300069.
KEYWORD
nonn,easy
AUTHOR
Thomas Zaslavsky, Mar 14 2010
EXTENSIONS
Information added to name and comments by Thomas Zaslavsky, Apr 24 2010
STATUS
approved
A300075 Period 6: repeat [0, 1, 1, 2, 2, 2]. +10
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, 0, 1, 1, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Underlying A300076.
LINKS
Table of n, a(n) for n=0..77.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
FORMULA
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
CROSSREFS
Cf. A300067, A300069, A300076.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 03 2018
STATUS
approved
A300290 Period 6: repeat [0, 1, 2, 2, 3, 3]. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Underlying A300068(n+2), n >= 0.
LINKS
Table of n, a(n) for n=0..89.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
FORMULA
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
CROSSREFS
Cf. A300067, A300069, A300068, A300075.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 03 2018
STATUS
approved
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)