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A322632
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Decimal expansion of the real solution to 23*x^5 - 41*x^4 + 10*x^3 - 6*x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.
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2
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1, 6, 3, 0, 2, 5, 7, 6, 6, 2, 9, 9, 0, 3, 5, 0, 1, 4, 0, 4, 2, 4, 8, 0, 1, 8, 4, 9, 3, 1, 5, 9, 8, 6, 3, 0, 0, 5, 1, 4, 5, 8, 4, 4, 2, 6, 6, 9, 0, 1, 4, 9, 4, 0, 5, 8, 4, 9, 8, 5, 0, 2, 6, 5, 9, 5, 2, 5, 6, 8, 9, 1, 2, 9, 8, 6, 8, 5, 0, 4, 7, 9, 8, 3, 4, 1, 3, 2, 4, 1
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OFFSET
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1,2
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COMMENTS
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In his 2014 lecture in Paris "Problems That Philippe (Flajolet) Would Have Loved" D. Knuth discussed as Problem 4 "Lattice Paths of Slope 2/5" and reported as an empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. For the meaning of A and B see A322631. The exact values of a (this sequence) and b (A322633) were found in 2016 by Banderier and Wallner.
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LINKS
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EXAMPLE
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1.6302576629903501404248018493159863005145844266901494058498502659525689...
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MAPLE
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evalf[100](solve(23*x^5-41*x^4+10*x^3-6*x^2-x-1=0, x)[1]); # Muniru A Asiru, Dec 21 2018
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MATHEMATICA
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RealDigits[Root[23#^5 - 41#^4 + 10#^3 - 6#^2 - # - 1&, 1], 10, 100][[1]] (* Jean-François Alcover, Dec 30 2018 *)
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PROG
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(PARI) solve(x=1, 2, 23*x^5-41*x^4+10*x^3-6*x^2-x-1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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