cp's OEIS Frontend

A079269 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(1+1/b(n-1)); sequence gives numerator of b(n).

%I A079269 #21 Aug 05 2025 18:20:34
%S A079269 1,3,21,861,1275141,2551762438701,9546380157472159016030421,
%T A079269 126857284256055227389078067834858327568823447932861
%N A079269 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(1+1/b(n-1)); sequence gives numerator of b(n).
%C A079269 The next term is too large to include.
%H A079269 Leroy Quet, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/pipermail/seqfan/2003-February/002173.html">a(m) =a(m-1)+1/(1+1/a(m-1))</a>, posting to SeqFan mailing list, Feb 15 2003.
%F A079269 Conjecture: a(m+1) = a(m)^2 + a(m)^3 /(2a(m-1)^2) - a(m)a(m-1)^2/2 for m >= 2. - _Leroy Quet_
%e A079269 The b sequence begins 1, 3/2, 21/10, 861/310, 1275141/363010, 2551762438701/594665194510, ... = a(n)/A079278.
%p A079269 b := proc(n) option remember; if n=1 then 1 else b(n-1)+1/(1+1/b(n-1)); fi; end;
%t A079269 nxt[n_]:=n+1/(1+1/n); Numerator/@Nest[Append[#,nxt[Last[#]]]&,{1},10]  (* _Harvey P. Dale_, Apr 21 2011 *)
%Y A079269 Cf. A079278, A080581, A080582, A080984, A080985.
%K A079269 nonn,frac
%O A079269 1,2
%A A079269 _N. J. A. Sloane_, Feb 16 2003