%I #17 Sep 14 2022 21:45:56

%S 1,9,3,7,5,8,2,11,4,6,22,88,44,66,77,33,99,55,101,909,111,191,121,181,

%T 131,171,141,161,151,252,262,242,272,232,282,222,292,212,393,313,494,

%U 323,383,333,373,343,363,353,454,464,444,474,434,484,424

%N Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

%C Obviously the sequence cannot contain 0.

%C It is easy to prove that the sequence is a permutation of the nonzero palindromes (in the sense that it contains each of them exactly once).

%H Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2022/09/sums-with-palindromes.html">Sums with palindromes</a>, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022

%o (PARI) A357044_first(n, U=[0], a=9)={vector(n,k, k=U[1]; while(is_A002113(a+k=A262038(k+1)) || setsearch(U, k), ); U=setunion(U,[a=k]); while(#U>1 && U[2]==A262038(U[1]+1), U=U[^1]); a)}

%o (Python)

%o from itertools import count, islice

%o def ispal(n): s = str(n); return s == s[::-1]

%o def nextpal(p): # next largest palindrome after palindrome p

%o d = str(p)

%o if set(d) == {'9'}: return int('1' + '0'*(len(d)-1) + '1')

%o h = str(int(d[:(len(d)+1)//2]) + 1)

%o return int(h + h[:-1][::-1]) if len(d)&1 else int(h + h[::-1])

%o def agen():

%o aset, pal, minpal = {1}, 1, 2

%o while True:

%o an = pal; yield an; aset.add(an); pal = minpal

%o while pal in aset or ispal(an+pal): pal = nextpal(pal)

%o while minpal in aset: minpal = nextpal(minpal)

%o print(list(islice(agen(), 55))) # _Michael S. Branicky_, Sep 14 2022

%Y Cf. A002113 (palindromes), A029742 (non-palindromes), A262038 (next palindrome), A357045 (non-palindromes with palindromic sum of neighbors).

%K nonn,base

%O 1,2

%A _Eric Angelini_ and _M. F. Hasler_, Sep 14 2022