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%I #74 Dec 20 2020 04:47:01
%S 9,0,999,9009,99999,906609,9999999,99000099,999969999,9966006699,
%T 99999999999,999000000999,9999998999999,99956644665999,
%U 999999999999999,9999000000009999,99999999799999999,999900665566009999,9999999997999999999,99999834000043899999,999999999999999999999,9999994020000204999999,99999999999899999999999
%N a(n) is the largest n-digit palindrome that is the product of two numbers having an equal number of digits, or 0 if no such palindrome exists.
%C From _Chai Wah Wu_, Sep 30 2019: (Start)
%C Note that the product decomposition satisfying the conditions is not necessarily unique. For instance, a(5) = 99999 = 369*271 = 123*813 and a(9) = 50001*19999 = 16667*59997.
%C When n is odd, a(n) in decimal are all 9's with the possible exception of the middle digit which can be 6,7,8 or 9, i.e. a(n) = 10^n-1-k*10^((n-1)/2) for some 0 <= k <= 3.
%C In particular, a(2m+1) >= (2*10^m-1)(5*10^m+1) = 10^(2m+1)-3*10^m-1. This inequality is an equality for m = 4, 15, 18, 20, 23, 29, 33, 34, 35.
%C See the "Decomposition of a(n) for odd n" file in the Links section for examples.
%C a(4m) >= (10^(2m)-1)(10^(2m)-10^m+1). The inequality is strict for m = 5. Is this a rare occurrence?
%C (End)
%C a(24) = 999999000001 * 999999999999 = 999999000000000000999999. - _David A. Corneth_, Sep 30 2019
%H Math StackExchange, <a href="https://math.stackexchange.com/questions/260698/proof-for-all-largest-palindrome-numbers-for-each-power-of-10">Proof for all largest palindrome numbers for each power of 10</a>
%H Donghwi Park, <a href="https://github.com/gwahak/mathematics/blob/master/A308803.ipynb">Source code for even n and all examples for n<=21</a>
%H Chai Wah Wu, <a href="/A308803/a308803_2.txt">Decomposition of a(n) for odd n</a>
%e a(1)=9 because 3*3=9;
%e a(2)=0 because there is no such palindrome;
%e a(3)=999 because 27*37=999;
%e a(4)=9009 because 99*91=9009;
%e a(5)=99999 because 369*271=99999;
%e a(6)=906609 because 993*913=906609;
%e a(7)=9999999 because 2151*4649=9999999;
%e a(8)=99000099 because 9999*9901=99000099;
%e a(9)=999969999 because 50001*19999=999969999;
%e a(10)=9966006699 because 99979*99681=9966006699;
%e a(11)=99999999999 because 194841*513239=99999999999;
%e a(12)=999000000999 because 999999*999001=999000000999;
%e a(13)=9999998999999 because 2893921*3455519=9999998999999.
%Y Cf. A327897. Subsequence of A002113.
%K nonn,base
%O 1,1
%A _Donghwi Park_, May 06 2019
%E a(14)-a(20) from _Jon E. Schoenfield_, May 10 2019
%E a(21) from _Donghwi Park_, Jul 16 2019
%E a(22)-a(23) from _Chai Wah Wu_, Sep 30 2019
%E a(20) corrected by _Donghwi Park_, Dec 18 2020
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