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%I #14 Sep 15 2022 11:48:42
%S 10,12,21,23,32,34,43,45,54,47,19,14,30,25,41,36,52,49,17,16,28,27,39,
%T 38,50,51,15,18,26,29,37,40,48,53,13,20,24,31,35,42,46,65,56,75,76,85,
%U 86,95,96,106,116,126,136,146,157,105,97,64,57,74
%N Lexicographically earliest sequence of distinct non-palindromic numbers (A029742) such that a(n)+a(n+1) is always a palindrome (A002113).
%C Conjecture: The sequence contains all non-palindromic numbers (A029742).
%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) A357045_first(n, U=[9], a=1)={vector(n, k, k=U[1]; until( is_A002113(a+k) && !is_A002113(k) && !setsearch(U, k), k++); U=setunion(U,[a=k]); while(#U>1 && U[2]==U[1]+1+is_A002113(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 agen():
%o aset, k, mink = {10}, 10, 12
%o while True:
%o an = k; yield an; aset.add(an); k = mink
%o while k in aset or ispal(k) or not ispal(an+k): k += 1
%o while mink in aset: mink += 1
%o print(list(islice(agen(), 60))) # _Michael S. Branicky_, Sep 14 2022
%Y Cf. A029742 (non-palindromes), A002113 (palindromes), A357044 (palindromes with non-palindromic sum of neighbors).
%K nonn,base
%O 1,1
%A _Eric Angelini_ and _M. F. Hasler_, Sep 14 2022
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