cp's OEIS Frontend

A307129 Lexicographically earliest sequence of distinct terms such that the sequence of digits has alternating parity, and the same holds for the digits of the sequence a(n) + a(n+1).

%I A307129 #23 Dec 19 2020 10:14:49
%S A307129 1,29,21,49,23,27,25,45,47,43,258,12,18,14,16,34,36,38,32,58,123,87,
%T A307129 214,56,125,85,216,54,127,83,218,52,129,81,2929,89,212,78,1014,76,
%U A307129 1016,74,1018,72,1218,1812,1238,1814,1216,1816,1214,1818,1212,1838,1232,1858,1234,1836,1236,1834,1256,3814,1258,1832,1418,1612,1438,1614,1416,1616,1414,1618,1412,1638,1432,1658,1434,1636,1436,1634,1456,3614,1458,1632,3418,1652,3438,1654,3416,1656,3414,1676,3616,1454,3618,1452,3638,1852,3218,1854
%N A307129 Lexicographically earliest sequence of distinct terms such that the sequence of digits has alternating parity, and the same holds for the digits of the sequence a(n) + a(n+1).
%C A307129 In the sequel we use S(n) = a(n) and T(n) = a(n) + a(n+1).
%C A307129 T is the sequence of the "first sums" of the successive pairs of S's adjacent terms. S(1) + S(2) = 1 + 29 = 30 and thus T(1) = 30; S(2) + S(3) = 29 + 21 = 50 and thus T(2) = 50; S(3) + S(4) = 21 + 49 = 70 and thus T(3) = 70; etc.
%C A307129 T has an entry for itself in the OEIS (A307130).
%C A307129 S and T need a lot of backtracking to be computed; this means that the last few terms of S and T might evolve. However, the first 100 terms proposed here seem correct.
%C A307129 It appears that the sequence can be computed in a greedy way, by discarding only the last term when it's impossible to find a successor for it. It also appears that the lexicographic earliest sequence following the same rules but starting with a(1) = 0 (or with a(1) = 2) is given, after this initial term, by the terms following S(11) = 258, i.e., a(2) = S(12) = 12, a(3) = S(13) = 18, etc. - _M. F. Hasler_, Apr 08 2019
%e A307129 The first terms of S are 1, 29, 21, 49, 23, 27, 25, 45, 47, 43, 258, ... and we see that the digits of S follow the pattern odd/even/odd/even...
%e A307129 The first terms of T are 30, 50, 70, 72, 50, 52, 70, 92, 90, 301, ... and we see that the digits of T follow the same odd/even/odd/even... pattern.
%o A307129 (PARI) okapi(N,a=1,U=[])={local(good(t)=if( t>T*98\99, T*=10; T*10\99+(t<11)*11, for(p=1,oo, t+=10^p; t\10^p%10>1 && return(t); t>=T&&break); (t+2*T*=10)\10),T, S,Sb,b=-1); while( N>#U=setunion(U,[a]), b!=a&& print1(a","); my(t=1-a%2); T=10; while((t+=2)%10>1 ||99*a+99>t=good(t), if( !setsearch(U,t) && setsearch([[1],[2]], Set(digits(fromdigits(concat(S, digits(a+t)%2),2),4))), Sb=S; S=2-(a+t)%2; b=a; a=t; next(2))); print1("no: "); S=Sb; a=b; N++);a} \\ 2nd & 3rd (optional) arg allow to specify the initial value and forbid specific values. - _M. F. Hasler_, Apr 08 2019
%Y A307129 Cf. A307130 which is the associated sequence T.
%Y A307129 See also: A097962, A098951.
%K A307129 base,nonn
%O A307129 1,2
%A A307129 _Eric Angelini_ and _Lars Blomberg_, Mar 26 2019
%E A307129 Edited by _M. F. Hasler_, Apr 08 2019