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A128531 a(n) = numerator of r(n): r(n) is such that the continued fraction (of rational terms) [r(1);r(2),...r(n)] equals the n-th Fibonacci number, for every positive integer n.

%I A128531 #16 Sep 24 2024 09:19:47
%S A128531 1,1,-2,3,-10,6,-65,378,-5525,16632,-1278485,25147584,-1012815817,
%T A128531 8022079296,-2114837334805,570081043090944,-60533314393713485,
%U A128531 1256458618972440576,-4540728540084435567025,1677888660820605842036736,-466914087740138106185288665
%N A128531 a(n) = numerator of r(n): r(n) is such that the continued fraction (of rational terms) [r(1);r(2),...r(n)] equals the n-th Fibonacci number, for every positive integer n.
%F A128531 For n>=4, r(n) = -F(n)/(F(n-3) r(n-1)), where F(n) is the n-th Fibonacci number.
%e A128531 The 5th Fibonacci number = 5 = 1 +1/(1 +1/(-2 +1/(3/2 -3/10))).
%e A128531 The 6th Fibonacci number = 8 = 1 +1/(1 +1/(-2 +1/(3/2 +1/(-10/3 +5/6)))).
%p A128531 L2cfrac := proc(L,targ) local a,i; a := targ ; for i from 1 to nops(L) do a := 1/(a-op(i,L)) ; od: end: A128531 := proc(nmax) local b,n,bnxt; b := [1] ; for n from nops(b)+1 to nmax do bnxt := L2cfrac(b,combinat[fibonacci](n+1)) ; b := [op(b),bnxt] ; od: [seq( numer(b[i]),i=1..nops(b))] ; end: A128531(22) ; # _R. J. Mathar_, Oct 09 2007
%t A128531 r[n_] := r[n] = Switch[n, 1, 1, 2, 1, 3, -2, _, -Fibonacci[n]/(Fibonacci[n-3]*r[n-1])];
%t A128531 a[n_] := Numerator[r[n]];
%t A128531 Table[a[n], {n, 1, 21}] (* _Jean-François Alcover_, Sep 24 2024 *)
%Y A128531 Cf. A128532.
%K A128531 frac,sign
%O A128531 1,3
%A A128531 _Leroy Quet_, Mar 08 2007
%E A128531 More terms from _R. J. Mathar_, Oct 09 2007