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A240601
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Recursive palindromes in base 10: palindromes n where each half of the digits of n is also a recursive palindrome.
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5
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999, 1111
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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A number n with m digits in base 10 is a member if n is a palindrome, and the first floor(m/2) digits of n is already a previous term of a(n). All repdigit numbers are terms of a(n). Fast generation of new terms with 2m digits can be done by concatenating the previous terms with m digits twice. Fast generation of new terms with 2m+1 digits can be done by concatenating the previous terms with m digits twice with any single digit in the middle. The smallest palindrome which is not a member of a(n) is 1001.
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LINKS
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EXAMPLE
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11011 is in the sequence since it is a palindrome of 5 digits, and the first floor(5/2) digits of it, 11, is also a term. 1001 and 10001 are not in the sequence since 10 is not in the sequence.
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PROG
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(Python)
from itertools import product
def pals(d, base=10): # all d-digit palindromes as strings
digits = "".join(str(i) for i in range(base))
for p in product(digits, repeat=d//2):
if d//2 > 0 and p[0] == "0": continue
left = "".join(p); right = left[::-1]
for mid in [[""], digits][d%2]: yield left + mid + right
def auptod(dd):
for d in range(1, dd+1):
for p in pals(d//2):
if d//2 == 0: p = ""
elif p[0] == "0": continue
for mid in [[""], "0123456789"][d%2]: yield int(p+mid+p[::-1])
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CROSSREFS
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KEYWORD
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base,nonn,nice
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AUTHOR
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STATUS
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approved
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