|
|
A049415
|
|
Number of squares (of positive integers) with n digits.
|
|
9
|
|
|
3, 6, 22, 68, 217, 683, 2163, 6837, 21623, 68377, 216228, 683772, 2162278, 6837722, 21622777, 68377223, 216227767, 683772233, 2162277661, 6837722339, 21622776602, 68377223398, 216227766017, 683772233983, 2162277660169
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Lim_{n->infinity} a(2n)/10^n = 1 - 1/sqrt(10);
|
|
LINKS
|
|
|
FORMULA
|
a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))).
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...;
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)
|
|
EXAMPLE
|
22 squares (100=10^2, 121=11^2, ...., 961=31^2) have 3 digits, hence a(3)=22.
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(Magma) [Ceiling(Sqrt(10^n))-Ceiling(Sqrt(10^(n-1))) : n in [1..30]]; // Vincenzo Librandi, Oct 01 2011
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,base
|
|
AUTHOR
|
Ulrich Schimke (ulrschimke(AT)aol.com)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|