%N Fibonacci-like sequence of numbers with nondecreasing positive digits. Let a^+ denote the number that is obtained from a if its positive digits are written in nondecreasing order, while zeros remain in their places. Let a<+>b = (a + b)^+. a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).
%C Note that operation n^+ differs from the one in A004185. If a term of the sequence has k digits, then it is followed by terms with >=k digits. The sequence has 7 terms with 1 digit, 13 terms with 2 digits, 30 terms with 3 digits, etc. The corresponding maximal terms are 8, 59, 559, etc.
%C The sequence is eventually periodic with period of length 144 and the first position of period 237. - _Peter J. C. Moses_, Feb 09 2014
%H Peter J. C. Moses, <a href="/A237568/b237568.txt">Table of n, a(n) for n = 0..952</a>
%t a:=0;a:=1;a[n_]:=a[n]=FromDigits[Insert[DeleteCases[Sort[#],0],0,1+#-Range[Length[#]]&[Position[#,0]]]&[IntegerDigits[a[n-1]+a[n-2]]]]; Map[a,Range[0,99]] (* _Peter J. C. Moses_, Feb 09 2014 *)
%Y Cf. A000045, A001129, A004185, A069638.
%A _Vladimir Shevelev_, Feb 09 2014