A331557 -id:A331557 - cp's OEIS frontend

A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

5, 9, 14, 99, 52, 89, 100, 407, 268, 10769, 10890, 99, 99, 4400, 8900, 9890, 10000, 97625, 1089, 3584, 99, 629882, 1099890, 10989, 926, 890000, 8491505, 10890099, 8229644, 9999989, 69923062, 10890000, 99099000, 43337905, 99990089, 962943454, 109890, 454649691

Offset: 1

James D. Klein, Jan 20 2020

When normalized over (0,1) by their respective palindrome-free interval about a 196-iterate, it has been empirically observed that the frequency distribution of this sequence appears to be quite symmetric about 0.5, as well as fractal when plotting the distribution over decreasing bin sizes.

The 196-iterates referred to here come from the reverse-and-add process generating A006960.

			The first term is 5 since 196-191 = 5
The second term is 9 since 887-878 = 9, etc.

Cf. A006960, A331557, A331560.

Map[Block[{k = # - 1}, While[k != IntegerReverse@ k, k--]; # - k] &, NestList[# + IntegerReverse[#] &, 196, 25]] (* brute force, or *)
Map[# - Block[{n = #, w, len, ww}, w = IntegerDigits[n]; len = Length@ w; ww = Take[w, Ceiling[len/2] ]; If[# < n, #, FromDigits@ Flatten@{#, If[OddQ@ len, Reverse@ Most@ #, Reverse@ #]} &@ If[Last@ ww == 0, MapAt[# - 1 &, Most@ ww, -1]~Join~{9}, MapAt[# - 1 &, ww, -1]]] &@ FromDigits@ Flatten@ {ww, If[OddQ@ len, Reverse@ Most@ ww, Reverse@ ww]}] &, NestList[# + IntegerReverse[#] &, 196, 37]] (* Michael De Vlieger, Jan 22 2020 *)

# Slow Brute-force
n = 196
while n < 10**15:
  m = n
  while m != int(str(m)[::-1]): m+=-1
  print(n-m, end=', ')
  n = n + int(str(n)[::-1])

a(n) = A331560(n) - A331557(n).

More terms from Michael De Vlieger, Jan 22 2020

A331560 Size of the palindrome-free intervals about the 196-iterates, A006960.

Original entry on oeis.org

11, 10, 110, 110, 100, 100, 110, 1100, 1000, 11000, 11000, 11000, 11000, 10000, 10000, 10000, 11000, 110000, 110000, 100000, 100000, 1100000, 1100000, 1100000, 1000000, 1000000, 11000000, 11000000, 10000000, 10000000, 110000000, 110000000, 110000000, 100000000

Offset: 1

James D. Klein, Jan 20 2020

By empirical observation, the integers in this sequence are of the form 10*10^n and 11*10^n, n >= 0, since they are the difference of consecutive palindromes surrounding the 196-iterates. (No differences of 2 observed.)

			191 < 196 < 202, 202 - 191 = 11;
878 < 887 < 888, 888 - 878 = 10; etc.

Cf. A006960, A331556, A331557.

# Palindrome-free interval about 196 offsets. Slow brute-force
n = 196
while n < 10**15:
     m = n
     while m != int(str(m)[::-1]): m+=1
     s = m
     m = n
     while m != int(str(m)[::-1]): m-=1
     print(s-m)
     n = n + int(str(n)[::-1])

a(n) = A331556(n) + A331557(n).

a(31)-a(34) from Jinyuan Wang, Feb 29 2020

cp's OEIS Frontend

A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

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