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A319477 Nonnegative integers which cannot be obtained by adding exactly two nonzero decimal palindromes.

%I A319477 #28 Jan 08 2025 11:13:18
%S A319477 0,1,21,32,43,54,65,76,87,98,111,131,141,151,161,171,181,191,201,1031,
%T A319477 1041,1042,1051,1052,1053,1061,1062,1063,1064,1071,1072,1073,1074,
%U A319477 1075,1081,1082,1083,1084,1085,1086,1091,1092,1093,1094,1095,1096,1097,1099
%N A319477 Nonnegative integers which cannot be obtained by adding exactly two nonzero decimal palindromes.
%C A319477 Every integer larger than two can be obtained by adding exactly three nonzero decimal palindromes.
%C A319477 The nonzero palindromes of this sequence are in A213879.
%H A319477 Alois P. Heinz, <a href="/A319477/b319477.txt">Table of n, a(n) for n = 1..65536</a>
%H A319477 Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://arxiv.org/abs/1602.06208">Every positive integer is a sum of three palindromes</a>, arXiv: 1602.06208 [math.NT], 2017, <a href="https://doi.org/10.1090/mcom/3221">Math. Comp.</a> 87 (2018), 3023-3055.
%H A319477 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=OKhacWQ2fCs">Every Number is the Sum of Three Palindromes</a>, Numberphile video (2018)
%F A319477 A319468(a(n)) = 0.
%p A319477 p:= proc(n) option remember; local i, s; s:= ""||n;
%p A319477       for i to iquo(length(s), 2) do if
%p A319477         s[i]<>s[-i] then return false fi od; true
%p A319477     end:
%p A319477 h:= proc(n) option remember; `if`(n<1, 0,
%p A319477      `if`(p(n), n, h(n-1)))
%p A319477     end:
%p A319477 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i<n,
%p A319477       0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p A319477     end:
%p A319477 g:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(2):
%p A319477 a:= proc(n) option remember; local j; for j from 1+
%p A319477       `if`(n=1, -1, a(n-1)) while g(j)<>0 do od; j
%p A319477     end:
%p A319477 seq(a(n), n=1..80);
%Y A319477 Cf. A002113, A035137 (allowing zero), A213879, A261131, A319453, A319468, A319586.
%K A319477 nonn,base
%O A319477 1,3
%A A319477 _Alois P. Heinz_, Sep 19 2018