A357044 Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

1, 9, 3, 7, 5, 8, 2, 11, 4, 6, 22, 88, 44, 66, 77, 33, 99, 55, 101, 909, 111, 191, 121, 181, 131, 171, 141, 161, 151, 252, 262, 242, 272, 232, 282, 222, 292, 212, 393, 313, 494, 323, 383, 333, 373, 343, 363, 353, 454, 464, 444, 474, 434, 484, 424

Offset: 1

Author

Eric Angelini and M. F. Hasler, Sep 14 2022

Keywords

• nonn
• base

Comments

Obviously the sequence cannot contain 0.

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).

Links

• Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022
• Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022 [Cached copy]

Crossrefs

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

Programs

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)}

Python

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def nextpal(p): # next largest palindrome after palindrome p
    d = str(p)
    if set(d) == {'9'}: return int('1' + '0'*(len(d)-1) + '1')
    h = str(int(d[:(len(d)+1)//2]) + 1)
    return int(h + h[:-1][::-1]) if len(d)&1 else int(h + h[::-1])
def agen():
    aset, pal, minpal = {1}, 1, 2
    while True:
        an = pal; yield an; aset.add(an); pal = minpal
        while pal in aset or ispal(an+pal): pal = nextpal(pal)
        while minpal in aset: minpal = nextpal(minpal)
print(list(islice(agen(), 55))) # Michael S. Branicky, Sep 14 2022