A322633 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

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 and Michael Wallner, Lattice paths of slope 2/5, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), SIAM, 2015, 105-113; arXiv, arXiv:1605.02967 [cs.DM], 2016.

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

Index entries for algebraic numbers, degree 5.

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

MATHEMATICA

RealDigits[(3/7) * Root[11571875*x^5 - 5363750*x^4 + 628250*x^3 - 97580*x^2 + 5180*x - 142, 1], 10, 120][[1]] (* Amiram Eldar, Apr 19 2026 *)

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.