A306486 Number of squares in the interval [e^(n-1), e^n).

0, 1, 1, 2, 3, 5, 8, 13, 21, 36, 58, 96, 159, 262, 431, 712, 1172, 1934, 3189, 5256, 8667, 14289, 23559, 38841, 64039, 105583, 174076, 287003, 473188, 780155, 1286258, 2120681, 3496412, 5764609, 9504233, 15669832, 25835185, 42595018, 70227313, 115785266

Offset: 0

Author

Alexei Kourbatov, Feb 18 2019

Keywords

• nonn

Comments

The lower endpoint e^(n-1) is included; the upper endpoint is not included. The terms a(0) to a(8) coincide with the Fibonacci numbers.

Examples

			Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4607 (first 501 terms from Alexei Kourbatov)

Crossrefs

Cf. A000045, A000290, A001113, A005181, A019774, A290506, A306604.

Programs

Maple

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(exp(i/2))):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 18 2019

PARI

a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
for(n=0,50,print1(a(n)", "))

Formula

a(n) = ceiling(sqrt(exp(n))) - ceiling(sqrt(exp(n-1))).

From Alois P. Heinz, Feb 19 2019: (Start)

Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774.

a(n) = A005181(n+1) - A005181(n). (End)

a(n) = (1-1/sqrt(e))*e^(n/2)+O(1) ~ 0.39346934...*e^(n/2) ~ A290506*e^(n/2). - Alexei Kourbatov, Feb 20 2019