A007567 Knopfmacher expansion of 1/2: a(n+1) = a(n-1)(a(n)+1)-1. (Formerly M2242)

-3, -2, 2, 5, 11, 59, 659, 38939, 25661459, 999231590939, 25641740502411581459, 25622037156669717708454796390939, 656993627914472375437286314449293585586011019581459, 16833515146119850260546015286782697097805280607642932235667159033564811666316390939

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

Table of n, a(n) for n=0..13.

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 = 1.438209999512701281674411567... . - Vaclav Kotesovec, Mar 06 2016

Mathematica

nxt[{a_, b_}]:={b, a(b+1)-1}; Join[{-3, -2}, Transpose[NestList[nxt, {2, 5}, 12]][[1]]] (* Harvey P. Dale, Oct 19 2012 *)
Flatten[{-3, -2, RecurrenceTable[{a[n+1] == a[n-1]*(a[n] + 1) - 1, a[1] == 2, a[2] == 2}, a, {n, 2, 14}]}] (* Vaclav Kotesovec, Mar 06 2016 *)

Crossrefs

Sequence in context: A021035 A371942 A259967 * A093055 A285733 A106335

Adjacent sequences: A007564 A007565 A007566 * A007568 A007569 A007570

Keyword

sign,easy,nice

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Christian G. Bower, Oct 15 1999

One additional term from Harvey P. Dale, Oct 19 2012

Status

approved