A101703 - OEIS

%I #15 Jun 21 2015 14:26:46

%S 21,291,885,2991,29991,234651,299991,2340651,2999991,8221845,23400651,

%T 29346591,29999991,234000651,293406591,299999991,2340000651,

%U 2346534651,2934006591,2993465991,2999999991,23400000651,23465934651,29340006591,29934065991,29999999991,82277815845,234000000651

%N Numbers n such that reversal(n) = (2/3)*n - 2.

%C 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.

%C 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.

%C Except for 885 & 8221845 all known terms of this sequence are of the form f(r,s,t,z).

%C 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).

%C 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.

%C From _David Wasserman_, Mar 27 2008: (Start)

%C 234653406534651 is a term that doesn't fit the f(r,s,t,z) format.

%C 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.

%C 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)

%H Giovanni Resta, <a href="/A101703/b101703.txt">Table of n, a(n) for n = 1..56</a> (terms < 10^14)

%e f(0,1,2,3) = 2934006534006534006591 is in the sequence because reversal(2934006534006534006591) = 1956004356004356004392 = (2/3)*2934006534006534006591-2.

%t Do[If[FromDigits[Reverse[IntegerDigits[n]]] == 2/3*n - 2, Print[n]], {n, 1100000000}]

%Y Cf. A086947, A069215, A101700, A101704.

%K base,nonn

%O 1,1

%A _Farideh Firoozbakht_, Dec 31 2004

%E More terms from _David Wasserman_, Mar 27 2008