A007568 Knopfmacher expansion of 2/3: a(n+1) = a(n-1)(a(n)+1)-1. (Formerly M3209)

4, 3, 3, 11, 35, 395, 13859, 5474699, 75873867299, 415386585433442699, 31516986660961757816520267299, 13091733472248265157675672660811061946713442699, 412611989213715130181341760777978491100613491478334525377331427960520267299

Offset

0,1

References

A. Knopfmacher, ``Rational numbers with predictable Engel product expansions,'' in G. E. Bergum et al., eds., Applications of Fibonacci Numbers. Vol. 5, pp. 421-427.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..17

Index entries for sequences related to Engel expansions

Formula

a(n) ~ c^(phi^n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.705023526853159146712304952... . - Vaclav Kotesovec, Mar 06 2016

Mathematica

t={1, 3}; Do[AppendTo[t, (t[[-1]]+1)*t[[-2]]-1], {n, 11}]; t[[1]]=4; t (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Flatten[{4, RecurrenceTable[{a[n+1] == a[n-1]*(a[n] + 1) - 1, a[1] == 3, a[2] == 3}, a, {n, 1, 12}]}] (* Vaclav Kotesovec, Mar 06 2016 *)

Crossrefs

Sequence in context: A055525 A370574 A309046 * A091884 A255257 A306769

Adjacent sequences: A007565 A007566 A007567 * A007569 A007570 A007571

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Status

approved