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A049415 Number of squares (of positive integers) with n digits.

%I A049415 #20 Sep 08 2022 08:44:58
%S A049415 3,6,22,68,217,683,2163,6837,21623,68377,216228,683772,2162278,
%T A049415 6837722,21622777,68377223,216227767,683772233,2162277661,6837722339,
%U A049415 21622776602,68377223398,216227766017,683772233983,2162277660169
%N A049415 Number of squares (of positive integers) with n digits.
%C A049415 a(n) + A180426(n) + A180429(n) + A180347(n) = A052268(n).
%C A049415 Lim_{n->infinity} a(2n)/10^n = 1 - 1/sqrt(10);
%C A049415 lim_{n->infinity} a(2n-1)/10^n = 1/sqrt(10) - 1/10. - _Robert G. Wilson v_, Aug 29 2012
%H A049415 Vincenzo Librandi, <a href="/A049415/b049415.txt">Table of n, a(n) for n = 1..1000</a>
%F A049415 a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))).
%F A049415 From _Jon E. Schoenfield_, Nov 30 2019: (Start)
%F A049415 a(2n) = floor(10^n * (1 - 1/sqrt(10))), so each even-indexed term a(2n) is given by the first n digits (after the decimal point) of 1 - 1/sqrt(10) = 0.68377223398316...;
%F A049415 a(2n-1) = ceiling(10^n * (1/sqrt(10) - 1/10)), so each odd-indexed term a(2n-1) is given by the first n digits (after the decimal point) of 1/sqrt(10) - 1/10 = 0.21622776601683..., plus 1. (End)
%e A049415 22 squares (100=10^2, 121=11^2, ...., 961=31^2) have 3 digits, hence a(3)=22.
%t A049415 f[n_] := Ceiling[Sqrt[10^n - 1]] - Ceiling[Sqrt[10^(n - 1)]]; f[1] = 3; Array[f, 24] (* _Robert G. Wilson v_, Aug 29 2012 *)
%o A049415 (Magma) [Ceiling(Sqrt(10^n))-Ceiling(Sqrt(10^(n-1))) : n in [1..30]]; // _Vincenzo Librandi_, Oct 01 2011
%Y A049415 A049415(n) = A017936(n+1) - A017936(n) = A049416(n+1) - A049416(n).
%Y A049415 Cf. A062940.
%K A049415 nonn,easy,base
%O A049415 1,1
%A A049415 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A049415 More terms from _Dean Hickerson_, Jul 10 2001