%I #12 Dec 18 2020 04:17:27
%S 10,1090,12,18,14,16,25,27,23,29,21,49,32,38,34,36,45,47,43,1007,54,
%T 1009,41,1029,52,1018,56,1005,58,1003,67,1014,69,1001,89,1120,81,1021,
%U 83,1023,85,1041,61,1043,63,1027,65,1016,74,1030,70,1032,72,1034,90,1010,92,1012,78,1050,50,1052,76,1070,30
%N Numbers with equal number of even and odd digits such that the sum a(n) + a(n+1) has an equal number of even and odd digits too.
%C Lexicographically earliest sequence of distinct positive terms with this property.
%C Numbers with an odd digit length cannot be in this sequence.
%e a(1) + a(2) = 10 + 1090 = 1100 (which has an equal number of even and odd digits);
%e a(2) + a(3) = 1090 + 12 = 1102 (idem);
%e a(3) + a(4) = 12 + 18 = 30 (idem);
%e a(4) + a(5) = 18 + 14 = 32 (idem); etc.
%o (Python)
%o def cond(i):
%o stri = str(i)
%o se = sum(1 for d in stri if d in "02468")
%o so = sum(1 for d in stri if d in "13579")
%o return se == so
%o def aupto(nn):
%o alst, used = [0], set()
%o for n in range(1, nn+1):
%o an = 10
%o while True:
%o while an in used: an += 1
%o if cond(an) and cond(an + alst[-1]):
%o alst.append(an); used.add(an); break
%o an += 1
%o return alst[1:] # use alst[n] for a(n)
%o print(aupto(56)) # _Michael S. Branicky_, Dec 14 2020
%Y Cf. A339715 (same idea, replacing addition by multiplication), A227870 (numbers with equal number of even and odd digits).
%K base,nonn
%O 1,1
%A _Eric Angelini_ and _Carole Dubois_, Dec 14 2020