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A007567 Knopfmacher expansion of 1/2: a(n+1) = a(n-1)(a(n)+1)-1.
(Formerly M2242)
0

%I M2242

%S -3,-2,2,5,11,59,659,38939,25661459,999231590939,25641740502411581459,

%T 25622037156669717708454796390939,

%U 656993627914472375437286314449293585586011019581459,16833515146119850260546015286782697097805280607642932235667159033564811666316390939

%N Knopfmacher expansion of 1/2: a(n+1) = a(n-1)(a(n)+1)-1.

%D 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.

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

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%F 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

%t nxt[{a_,b_}]:={b,a(b+1)-1}; Join[{-3,-2},Transpose[NestList[nxt,{2,5},12]][[1]]] (* _Harvey P. Dale_, Oct 19 2012 *)

%t 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 *)

%K sign,easy,nice

%O 0,1

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E More terms from _Christian G. Bower_, Oct 15 1999

%E One additional term from _Harvey P. Dale_, Oct 19 2012

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Last modified February 16 08:26 EST 2019. Contains 320159 sequences. (Running on oeis4.)