A322632 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.

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

Offset: 1

Author

Hugo Pfoertner, Dec 21 2018

Keywords

• nonn
• cons

Comments

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.

Examples

			1.6302576629903501404248018493159863005145844266901494058498502659525689...
		

Links

• Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.
• D. E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.

Crossrefs

Cf. A322631, A322633.

Programs

Maple

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

Mathematica

RealDigits[Root[23#^5 - 41#^4 + 10#^3 - 6#^2 - # - 1&, 1], 10, 100][[1]] (* Jean-François Alcover, Dec 30 2018 *)

PARI

solve(x=1,2,23*x^5-41*x^4+10*x^3-6*x^2-x-1)