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%I #9 May 30 2019 08:59:40
%S 4,8,11,12,19,24,28,33,36,40,47,52,56,57,61,68,72,80,84,85,92,97,99,
%T 104,108,109,113,120,125,141,145,156,168,170,171,172,183,193,204,208,
%U 216,218,229,240,244,245,250,252,255,257,269,276,278,280,291,297,301,312
%N Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 3.
%C All the palindromic numbers in base 3 with an even number of digits belong to the sequence.
%C Here the fulcrum is between two digits while in the sequence from A282107 to A282115 is one of the digits.
%H Paolo P. Lava, <a href="/A282144/b282144.txt">Table of n, a(n) for n = 1..10000</a>
%e 229 in base 3 is 22111. If we split the number in 22 and 111 we have 2*1 + 2*2 = 6 for the left side and 1*1 + 1*2 + 1*3 = 6 for the right one.
%p P:=proc(n,h) local a,j,k: a:=convert(n, base, h):
%p for k from 1 to nops(a)-1 do
%p if add(a[j]*(k-j+1),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a))
%p then RETURN(n); break: fi: od: end: seq(P(i,3),i=1..10^3);
%Y Cf. A282107 - A282115, A282143, A282145 - A282151.
%K nonn,base,easy
%O 1,1
%A _Paolo P. Lava_, _Giovanni Resta_, Feb 07 2017