%I
%S 0,1,9,2,8,3,7,4,6,5,10,11,20,21,30,31,40,41,50,51,49,12,19,22,29,32,
%T 39,42,58,13,18,23,28,33,38,43,48,52,53,47,14,17,24,27,34,37,44,56,15,
%U 16,25,26,35,36,45,46,54,55,57,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80
%N The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.
%C The sequence starts with a(0) = 0 and is always extended with the smallest integer not yet present that does not lead to a contradiction. The sequence is a permutation of the natural numbers.
%C Originally the sequence was defined starting with a(1) = 1 and using only positive integers. This leads to the same sequence restricted to positive indices, which yields a permutation of the positive integers.  _M. F. Hasler_, Feb 28 2018
%H M. F. Hasler, <a href="/A299957/b299957.txt">Table of n, a(n) for n = 0..10000</a>
%e 1 + 9 = 10; 9 + 2 = 11; 2 + 8 = 10; 8 + 3 = 11; 3 + 7 = 10; 7 + 4 = 11; 4 + 6 = 10; 6 + 5 = 11; etc.
%t Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[k + #[[1]], 10, 1] > 0], k++]; k]] &, {1}, 98] (* _Michael De Vlieger_, Feb 22 2018 *)
%o (PARI) a(n, f=1, a=0, u=[a])={for(n=a+1, n, f&&if(f==1,print1(a","),write(f,n1," "a)); for(k=u[1]+1, oo, setsearch(u, k)&&next; setsearch(Set(digits(a+k)),1)&&(a=k)&&break); u=setunion(u, [a]); u[2]==u[1]+1&&u=u[^1]); a} \\ _M. F. Hasler_, Feb 22 2018
%Y Cf. A299952 (different constraint: a(n) + a(n+1) must be substring of concatenation of a(1..n+1)).
%Y Cf. A299970, A299982, ..., A299988, A299969 (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).
%Y Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.
%K nonn,base
%O 0,3
%A _Eric Angelini_, Feb 22 2018
%E Extended to a(0) = 0 by _M. F. Hasler_, Feb 28 2018
