A369988 Decimal expansion of Mallows's constant or stribolic constant kappa (of order 1).

2, 7, 8, 8, 7, 7, 0, 6, 1

Offset: 0

Author

Roland Miyamoto, Feb 07 2024

Keywords

• nonn
• cons
• more

Comments

This constant is the area under the unique bijective, differentiable function h:[0,1]->[0,1] satisfying -c*h' = h^{-1} (compositional inverse) for some c > 0. That is, kappa = Integral_{t=0..1} h(t) dt, and then we also have kappa = c = -1/h'(0).

Equivalently, 1/kappa = 3.5858... is the only a > 0 such that there exists a differentiable function g:[0,a]->[0,a] which becomes its own derivative when rotated 90 degrees clockwise about the origin (into the fourth quadrant; whence the names "stribola" for g and h and "stribolic constant" for kappa, from Greek stribo=turn/twist), namely g(x):=h(kappa*x)/kappa for 0 <= x <= a = 1/kappa.

In 1997, Colin Mallows observed and conjectured that the rows in Levine's triangle A012257 take on stribolic shape and that A011784(n+1)/(A011784(n)*A011784(n-1)) converges as n->oo. Presuming his conjecture, the limit would equal kappa, while Mallows estimated it to be "approximately ... 0.277", see A011784. Later, in 2006, Martin Fuller suggested 0.27887706... for the limit, based on a numerical iteration, see A012257.

Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See first arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we could apply the transformation G four times and would obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.

It is an open question whether kappa is rational or irrational, algebraic or transcendental.

Examples

			0.278877061...
		

References

• N. J. A. Sloane, My Favorite Integer Sequences, in: C. Ding, T. Helleseth, H. Niederreiter (editors), Sequences and their Applications, Discrete Mathematics and Theoretical Computer Science, Springer, London (1999) 103-130.

Links

• Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.
• Roland Miyamoto, Solution to the iterative differential equation -gamma*g' = g^{-1}, arXiv:2404.11455 [math.CA], 2024.
• Roland Miyamoto and J. W. Sander, Solving the iterative differential equation -gamma*g' = g^{-1}, in: H. Maier, J. & R. Steuding (eds.), Number Theory in Memory of Eduard Wirsing, Springer, 2023, pp. 223-236.
• N. J. A. Sloane, My favorite integer sequences, in: Sequences and their Applications (Proceedings of SETA '98).
• N. J. A. Sloane, Colin Mallows, and Bjorn Poonen, Discussion of A011784. [Scans of pages 150-155 and 164 of Sloane's notebook "Lattices 77", from June-July 1997.]

Crossrefs

Cf. A369990, A369991, A369992, A369993, A011784, A012257, A014621.

Formula

Set kappa_n := A369990(n) / A369991(n). Then

(1) kappa = lim_{n->oo}kappa_n = inf{kappa_n: n >= 0},

(2) kappa_n - 1 + kappa_n/kappa_{n-1} < kappa < kappa_n for n=1,2,...,

(3) 0.2788770612338 < kappa_{23} - 1 + kappa_{23}/kappa_{22} < kappa < kappa_{23} < 0.2788770613941.