A322633 Decimal expansion of the real solution to 11571875*x^5 - 5363750*x^4 + 628250*x^3 - 97580*x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

1, 5, 8, 6, 6, 8, 2, 2, 6, 9, 7, 2, 0, 2, 2, 7, 7, 5, 5, 1, 4, 7, 8, 0, 4, 1, 9, 8, 1, 6, 3, 9, 2, 7, 3, 9, 2, 0, 6, 3, 7, 3, 7, 5, 7, 5, 5, 3, 0, 6, 6, 9, 7, 7, 9, 6, 3, 6, 6, 5, 2, 4, 0, 9, 5, 4, 4, 0, 7, 6, 7, 0, 2, 1, 0, 9, 5, 3, 1, 2, 5, 7, 9, 6, 1, 9, 1, 7, 1, 9, 2

Offset

0,2

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 (A322632) and b (this sequence) were found in 2016 by Banderier and Wallner.

Links

Table of n, a(n) for n=0..90.

Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.

Don E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.

Example

0.1586682269720227755147804198163927392063737575530669779636652409544...

Maple

evalf[100]((3/7)*solve(11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142=0, x)[1]); # Muniru A Asiru, Dec 21 2018

Prog

(PARI) (3/7)*solve(x=0, 1, 11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142)

Crossrefs

Cf. A322631, A322632.

Sequence in context: A301862 A245944 A160043 * A346443 A145432 A335565

Adjacent sequences: A322630 A322631 A322632 * A322634 A322635 A322636

Keyword

nonn,cons

Author

Hugo Pfoertner, Dec 21 2018

Status

approved