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A275448
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The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.
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1
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1, 2, 3, 4, 6, 12, 28, 65, 146, 327, 749, 1756, 4165, 9913, 23652, 56687, 136627, 330969, 804915, 1963830, 4805523, 11793046, 29019930, 71589861, 177006752, 438561959, 1088714711, 2707615555, 6745272783, 16830750107, 42058592797, 105248042792
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OFFSET
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2,2
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LINKS
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FORMULA
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G.f.: g(z) = (1-3z+3z^2 - Q)/(2z(1-z)), where Q = sqrt((1-3z+z^2)(1-3z+5z^2-4z^3)).
D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(7*n-11)*a(n-2) +(-37*n+107)*a(n-3) +3*(13*n-54)*a(n-4) +3*(-7*n+37)*a(n-5) +2*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 22 2022
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EXAMPLE
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a(4)=3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that only [1,1,1],[2,2], and [3] lead to weakly alternating bargraphs.
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MAPLE
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g := ((1-3*z+3*z^2-sqrt((1-3*z+z^2)*(1-3*z+5*z^2-4*z^3)))*(1/2))/(z*(1-z)): gser:= series(g, z=0, 43): seq(coeff(gser, z, n), n=2..40);
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MATHEMATICA
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terms = 32;
g[z_] = ((1 - 3z + 3z^2 - Sqrt[(1 - 3z + z^2)(1 - 3z + 5z^2 - 4z^3)])*(1/2) )/(z(1-z));
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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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