A306604 Number of perfect squares in the half-open interval [Pi^(n-1), Pi^n).

0, 1, 2, 2, 4, 8, 14, 23, 43, 75, 134, 236, 419, 743, 1316, 2333, 4135, 7329, 12992, 23026, 40813, 72338, 128218, 227259, 402806, 713955, 1265453, 2242956, 3975538, 7046456, 12489518, 22137096, 39236979, 69545736, 123266607, 218484372, 387253468, 686388899

Offset: 0

Author

Alois P. Heinz, Feb 27 2019

Keywords

• nonn

Comments

Inspired by A306486.

Examples

			a(4) = 4: in the interval [Pi^3, Pi^4) = [31.006..., 97.409...) = are four perfect squares: 36, 49, 64, 81.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4024

Crossrefs

Cf. A000290, A000796, A002161, A091925, A092425, A102475, A102477, A306486.

Programs

Maple

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(Pi^(i/2))):
seq(a(n), n=0..42);

Formula

a(n) = ceiling(Pi^(n/2)) - ceiling(Pi^((n-1)/2)).

a(n) = A102477(n) - A102477(n-1).

Sum_{i=0..n} a(i) = A102475(n) for n > 0.

Lim_{n->oo} a(n+1)/a(n) = sqrt(Pi) = 1.7724538509... = A002161.