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%I #23 Nov 24 2015 12:40:36
%S 1,6,4,9,61,504,100,489,2944,656,3844,34449,85636,516,961,6201,5625,
%T 43524,36729,7225,344,569,2996,361,201,64,1601,6004,7001,316,84,681,
%U 21,16,81,225,625,5001,3184,3449,2129,8225,424,36,481,636,804,609,1024,144
%N a(1) = 1, a(n) = smallest positive number not yet in the sequence such that the concatenation of a(n-1) and a(n) is a square.
%C For any x > 0, if d is large enough there are squares between 10^d*x + 10^(d-1) and 10^d*x + 10^d - 1. Thus the sequence is infinite.
%C a(3) = 4 is the minimum value of a(n) for n > 1. - _Altug Alkan_, Nov 24 2015 (This is because no square can end in 2 or 3, so 2 and 3 can never appear in the sequence. - _N. J. A. Sloane_, Nov 24 2015)
%H Robert Israel, <a href="/A264770/b264770.txt">Table of n, a(n) for n = 1..10000</a>
%e For n = 6, a(n-1) = 61. There are no squares of the form 61x or 61xy with x>=1. The least square of the form 61xyz with x >= 1 is 61504, and 504 has not appeared previously so a(6) = 504.
%p S:= {1};
%p A[1]:= 1;
%p for n from 2 to 100 do
%p found:= false;
%p x:= A[n-1];
%p for d from 1 while not found do
%p a:= ceil(sqrt(10^d*x +10^(d-1)));
%p b:= floor(sqrt(10^d*x + 10^d - 1));
%p Q:= map(t -> t^2 - 10^d*x, {$a..b}) minus S;
%p if nops(Q) >= 1 then
%p A[n]:= min(Q);
%p S:= S union {A[n]};
%p found:= true;
%p fi
%p od
%p od:
%p seq(A[n],n=1..100);
%t (*to get B numbers of the sequence*) A={1};i=1;While[i<B,i++;m=Last[A];d=0;flag=0;While[flag==0,d++;g0=Ceiling[Sqrt[m*10^d+10^(d-1)]];h=(m+1)10^d;a=g0;Label[L$];If[a^2<h,b=a^2-m*10^d;If[MemberQ[A,b],a++;Goto[L$],flag=1;AppendTo[A,b]]]]];A (* _Emmanuel Vantieghem_, Nov 24 2015 *)
%o (PARI) A264770(n,show=0,a=1,u=[])={for(n=2, n, u=setunion(u,[a]); show&&print1(a", "); my(k=3); until(!setsearch(u, k++) && issquare(eval(Str(a,k))),);a=k); a} \\ Use optional 2nd, 3rd or 4th argument to print intermediate terms, use another starting value, or exclude some numbers. - _M. F. Hasler_, Nov 24 2015
%Y Cf. A082209, A090566.
%K nonn,base
%O 1,2
%A _Robert Israel_, Nov 24 2015, following a suggestion from _N. J. A. Sloane_.
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