A366198 Any a(n) replacing the first digit of a(n+1) forms a palindrome. This is the lexicographically earliest sequence of distinct nonnegative integers with this property.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 11, 21, 12, 31, 13, 41, 14, 51, 15, 61, 16, 71, 17, 81, 18, 91, 29, 22, 32, 23, 42, 24, 52, 25, 62, 26, 72, 27, 82, 28, 92, 39, 33, 43, 34, 53, 35, 63, 36, 73, 37, 83, 38, 93, 49, 44, 54, 45, 64, 46, 74, 47, 84, 48, 94, 59, 55, 65, 56, 75, 57, 85, 58, 95

Offset: 1

Author

Eric Angelini, Oct 03 2023

Keywords

• base
• nonn

Comments

No integer > 1 ending in zero will appear in the sequence.

For n >= 10 the concatenation of a(n) and A217657(a(n+1)) is a palindrome.

Examples

			a(9)  =  8 replacing the first digit of a(10) =  9 forms   8, a palindrome;
a(10) =  9 replacing the first digit of a(11) = 19 forms  99, a palindrome;
a(11) = 19 replacing the first digit of a(12) = 11 forms 191, a palindrome;
a(12) = 11 replacing the first digit of a(13) = 21 forms 111, a palindrome;
a(13) = 21 replacing the first digit of a(14) = 12 forms 212, a palindrome; etc.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A082216, A217657, A318486.

Programs

Mathematica

terms=75; b[0]=0;
b[n_]:=b[n]=(k=1; While[MemberQ[Array[b,n-1],k]||!PalindromeQ[FromDigits@Flatten@ReplacePart[IntegerDigits@k,1-> IntegerDigits@b[n-1]]],k++]; k); t=0;While[Length[a=Join[Range[0,9],Flatten@Table[FromDigits@Flatten@Insert[#,Table[9,i],-2]&/@(IntegerDigits/@Array[b,9^2,10]),{i,0,t++}]]]Giorgos Kalogeropoulos, Oct 04 2023 *)

Python

from itertools import count, islice
def ispal(n): return (s:=str(n))==s[::-1]
def agen(): # generator of terms
    an, seen = 0, set()
    while True:
        yield an; seen.add(an); s = str(an)
        an = next(k for k in count(0) if k not in seen and ispal(s+str(k)[1:]))
print(list(islice(agen(), 80))) # Michael S. Branicky, Oct 04 2023

Python

# faster version suitable for generating b-file
from sympy import isprime
from itertools import count, islice, product
def pals(digs):
    yield from digs
    for d in count(2):
        for p in product(digs, repeat=d//2):
            left = "".join(p)
            for mid in [[""], digs][d%2]:
                yield left + mid + left[::-1]
def folds(s): # generator of suffixes of palindromes starting with s
    for i in range((len(s)+1)//2, len(s)+1):
        for mid in [True, False]:
            t = s[:i] + (s[:i-1][::-1] if mid else s[:i][::-1])
            if t.startswith(s):
                yield t[len(s):]
    yield from ("".join(p)+s[::-1] for p in pals("0123456789"))
def agen():
    s, seen = "0", {"0"}
    while True:
        yield int(s)
        found = False
        for end in folds(s):
            for start in "123456789":
                t = start + end
                if t not in seen:
                    found = True; break
            if found: break
        s, seen = t, seen | {t}
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 04 2023

Formula

For n >= 92, a(n) = 10*a(n-81) + 90 - 9*(a(n-81) mod 10). - David A. Corneth, Oct 04 2023