Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-30T22:08:03.582Z Has data issue: false hasContentIssue false
  • English
  • Français

A quadratic recurrence of Faltung type

Published online by Cambridge University Press:  24 October 2008

E. M. Wright
E. M. Wright
Affiliation:
University of Aberdeen
Get access Rights & Permissions [Opens in a new window]

Extract

We write and x1 = 1. In a recent paper (3), Stein and Everett consider the sequence defined by

and investigate whether

tends to a limit as n → ∞. The case b = 0 has a combinatorial interpretation (see (1)) and, in (2), they use this to prove that xn → e−1 in this case. Even for positive integral b, the number Sn has no known combinatorial interpretation, but they prove (3) that xn → x under the hypothesis that Sn+1Sn−1 ≥ Sn i.e. that Sn is convex. By computation and induction, they prove this hypothesis for b = k/5, where k = 1, 2,…, 50.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 2 , September 1980 , pp. 193 - 197
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Stein, P. R.On a class of linked diagrams. I. Enumeration. J. Combinatorial Theory A 24 (1978), 357–366; Math. Zbl. 395 (1979), 26–27.Google Scholar
(2)Stein, P. R. and Everett, C. J.On a class of linked diagrams. II. Asymptotics. Discrete Math. 22 (1978), 309–318; Math. Zbl. 395 (1979), 27.Google Scholar
(3)Stein, P. R. and Everett, C. J.On a quadratic recurrence rule of Faltung type. J. Combinatorial Inf. Syst. Sci. 3 (1978), 1–10; Math. Zbl. 393 (1979), 359.Google Scholar
(4)Wright, E. M.Formula for the number of labelled, connected, sparsely-edged graphs. Graph Theory Newsletter 6 (1977), no. 5, 6.Google Scholar
(5)Wright, E. M.Asymptotic formulae for the number of labelled, connected, sparsely-edged graphs. Graph Theory Newsletter 8 (1979), no. 6, 2.Google Scholar