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A321195
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Minimum number of monochromatic Schur triples over all 2-colorings of [n].
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1
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0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 33, 36, 40, 44, 48, 52, 57, 61, 66, 71, 76, 82, 87, 93, 99, 105, 111, 118, 124, 131, 138, 145, 153, 160, 168, 176, 184, 192, 201, 209, 218, 227, 236, 246, 255, 265, 275, 285, 295, 306, 316, 327, 338
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OFFSET
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1,7
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COMMENTS
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A Schur triple is an integer triple (x,y,x+y). A 2-coloring on [n] = {1,...,n} is a surjective map [n] --> {Red, Blue}. A triple (x,y,z) is monochromatic if the set {x,y,z} is mapped to one color. a(n) is exactly the minimum number of monochromatic Schur triples over all possible 2-colorings of [n]. The "optimal" coloring for all n (i.e. the coloring that produces the minimum number of monochromatic Schur triples) was shown by Robertson and Zeilberger in 1998 and later with a greedy algorithm by Thanatipanonda in 2009 (see the example for the optimal coloring of [11]). The exact formula for a(n) is computed based on this optimal coloring. One might notice in the literature that there is an additional assumption x<=y. We do not impose this constraint.
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LINKS
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C. Koutschan and E. Wong, Exact Lower Bounds for Monochromatic Schur Triples and Generalizations, In: Pillwein V., Schneider C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria), Springer, pp.223-248 (2020).
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1).
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FORMULA
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a(n) = floor((n^2 - 4*n + 6)/11).
G.f.: (-x^5 + x^6 - x^7 - x^10 + x^11 - x^12)/ (-1 + 2*x - x^2 + x^11 - 2*x^12 + x^13).
a(n) = 2*a(n-1) - a(n-2) + a(n-11) - 2*a(n-12) + a(n-13) for n > 13. - Colin Barker, Dec 18 2018
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EXAMPLE
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On the interval [11], the optimal coloring is R,R,R,R,B,B,B,B,B,B,R, which produces a minimum 7 monochromatic triples (x,y,x+y): (1,1,2), (1,2,3), (2,1,3), (1,3,4), (2,2,4), (3,1,4), (5,5,10).
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MAPLE
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seq(floor((n^2-4*n+6)/11), n=1..1000);
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MATHEMATICA
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Table[Floor[(n^2-4n+6)/11], {n, 1000}]
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PROG
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(GAP) List([1..100], n->Int((n^2-4*n+6)/11)); # Muniru A Asiru, Dec 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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