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 A321195 Minimum number of monochromatic Schur triples over all 2-colorings of [n]. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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 ). 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. LINKS Elaine Wong, Table of n, a(n) for n = 1..1000 B. Datskovsky, On the number of monochromatic Schur triples, Advances in Applied Math, 31 (2003), 193-198. A. Robertson and D. Zeilberger, A 2-coloring of [1,n] can have n^2/22+O(n) monochromatic Schur triples, but not less!, Electronic Journal of Combinatorics, 5 #R19 (1998). T. Schoen, The number of monochromatic Schur Triples, European Journal of Combinatorics, 20 (1999), 855-866. T. Thanatipanonda, On the Monochromatic Schur Triples Type Problem, Electronic Journal of Combinatorics, 16(1) #R14 (2009). Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1). FORMULA 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 EXAMPLE On the interval , 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). MAPLE seq(floor((n^2-4*n+6)/11), n=1..1000); MATHEMATICA Table[Floor[(n^2-4n+6)/11], {n, 1000}] PROG (GAP) List([1..100], n->Int((n^2-4*n+6)/11)); # Muniru A Asiru, Dec 19 2018 CROSSREFS Sequence in context: A011865 A085680 A253186 * A249020 A258470 A248357 Adjacent sequences:  A321192 A321193 A321194 * A321196 A321197 A321198 KEYWORD nonn AUTHOR Elaine Wong, Dec 18 2018 STATUS approved

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Last modified March 25 12:50 EDT 2019. Contains 321470 sequences. (Running on oeis4.)