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COMMENTS
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Numbers of the form 3*(10^n-3) are in the sequence, so A086947 is an infinite subsequence of this sequence. Also A101700 is a subsequence of this sequence.
Let f(r,s,t,z) = 2.(9)(r+s).(34.(0)(t).65)(z).(9)(s).1 where the dot between numbers means concatenation and "(m)(n)" means number of m's is n, for example f(0,2,1,3)= 299340653406534065991, it is interesting that all numbers of the form f(r,s,t,z) where r, s, t & z are nonnegative integers and r*z=0 are in this sequence.
Except for 885 & 8221845 all known terms of this sequence are of the form f(r,s,t,z).
For all r, s & t we have f(r,s,t,0)=f(r,s,0,0)=f(r+2s,0,0,0)=A086947(r+2s+1)= 3*(10^(r+2s+1)-3).
a(1) = 21 = f(0,0,0,0), a(2) = 291 = f(1,0,0,0), a(4) = 2991 = f(2,0,0,0) = f(0,1,0,0), a(5) = 29991 = f(3,0,0,0) = f(1,1,0,0), a(6) = 234651 = f(0,0,0,1), a(7) = 299991 = f(4,0,0,0) = f(0,2,0,0), a(8) = 2340651 = f(0,0,1,1), etc. Next term is greater than 11*10^8.
234653406534651 is a term that doesn't fit the f(r,s,t,z) format.
We may redefine f so that t is a vector of length z, which must be symmetrical to produce a member. For example f(0,0,[0,1,0],3) = 234653406534651 is a member, but f(0,0,[1,0,0],3) = 234065346534651 is not a member.
23465934651 is another member that doesn't fit the pattern. In general there may be any number of 9's between a 5 and a 3, provided that the 9's are symmetrical. So 2346593465934651 is a member, but 23465993465934651 is not. (End)
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