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A306486
Number of squares in the interval [e^(n-1), e^n).
2
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
0,4
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4607 (first 501 terms from Alexei Kourbatov)
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
EXAMPLE
Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.
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
PROG
(PARI) a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
for(n=0, 50, print1(a(n)", "))
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Feb 18 2019
STATUS
approved