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%I #34 Mar 19 2023 03:35:01
%S 5,9,14,99,52,89,100,407,268,10769,10890,99,99,4400,8900,9890,10000,
%T 97625,1089,3584,99,629882,1099890,10989,926,890000,8491505,10890099,
%U 8229644,9999989,69923062,10890000,99099000,43337905,99990089,962943454,109890,454649691
%N The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.
%C 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.
%C The 196-iterates referred to here come from the reverse-and-add process generating A006960.
%F a(n) = A331560(n) - A331557(n).
%e The first term is 5 since 196-191 = 5
%e The second term is 9 since 887-878 = 9, etc.
%t Map[Block[{k = # - 1}, While[k != IntegerReverse@ k, k--]; # - k] &, NestList[# + IntegerReverse[#] &, 196, 25]] (* brute force, or *)
%t 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 *)
%o (Python)
%o # Slow Brute-force
%o n = 196
%o while n < 10**15:
%o m = n
%o while m != int(str(m)[::-1]): m+=-1
%o print(n-m, end=', ')
%o n = n + int(str(n)[::-1])
%Y Cf. A006960, A331557, A331560.
%K nonn,base
%O 1,1
%A _James D. Klein_, Jan 20 2020
%E More terms from _Michael De Vlieger_, Jan 22 2020
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