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A181756 a(1) = 1, a(2) = 10. For n >= 3, a(n) is found by concatenating the first n-1 terms of the sequence and then dividing the resulting number by a(n-1).

%I A181756 #10 Oct 15 2024 04:23:34
%S A181756 1,10,11,1001,110001,1001000001,1100010000000001,
%T A181756 10010000010000000000000001,
%U A181756 110001000000000100000000000000000000000001,10010000010000000000000001000000000000000000000000000000000000000001
%N A181756 a(1) = 1, a(2) = 10. For n >= 3, a(n) is found by concatenating the first n-1 terms of the sequence and then dividing the resulting number by a(n-1).
%C A181756 The calculations for the first few values of the sequence are
%C A181756 ... a(3) = 110/10 = 11
%C A181756 ... a(4) = 11011/11 = 1001
%C A181756 ... a(5) = 110111001/1001 = 110001.
%C A181756 The above calculations are in base 10, but we get exactly the same results when working in an arbitrary base b. For example, with starting values a(1) = 1, a(2) = b (= 10 in base b), we get
%C A181756 ... a(3) = (b^2+b)/b = b+1 which equals 11 in base b,
%C A181756 ... a(4) = (b^4+b^3+b+1)/(b+1) = b^3+1 = 1001 in base b,
%C A181756 and so on.
%C A181756 For similarly defined sequences see A181754, A181755 and A181864 through A181870.
%F A181756 DEFINITION
%F A181756 a(1) = 1, a(2) = 10, and for n >= 3
%F A181756 (1)... a(n) = concatenate(a(1),a(2),...,a(n-1))/a(n-1).
%F A181756 RECURRENCE RELATION
%F A181756 For n >= 2
%F A181756 (2)... a(n+2) = 100^F(n)*a(n)+1,
%F A181756 where F(n) = A000045(n) are the Fibonacci numbers.
%F A181756 For n >= 2, a(n) has 2*F(n-1) digits.
%p A181756 #A181756
%p A181756 M:=10:
%p A181756 a:=array(1..M):s:=array(1..M):
%p A181756 a[1]:=1:a[2]:=10:
%p A181756 s[1]:=convert(a[1],string):
%p A181756 s[2]:=cat(s[1],convert(a[2],string)):
%p A181756 for n from 3 to M do
%p A181756 a[n] := parse(s[n-1])/a[n-1];
%p A181756 s[n]:= cat(s[n-1],convert(a[n],string));
%p A181756 end do:
%p A181756 seq(a[n],n = 1..M);
%Y A181756 Cf. A000045, A181754, A181755, A181864, A181865, A181866, A181867, A181868, A181869, A181870
%K A181756 easy,nonn,base
%O A181756 1,2
%A A181756 _Peter Bala_, Nov 09 2010