A005181 -id:A005181 - cp's OEIS frontend

A096766 Difference between ceiling(e^(n/2 - 1)) (A005181) and the n-th Fibonacci number (A000045).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 12, 27, 56, 110, 212, 397, 734, 1339, 2414, 4316, 7659, 13507, 23691, 41362, 71920, 124603, 215188, 370565, 636491, 1090709, 1865121, 3183224, 5423255, 9224601, 15666968, 26571801, 45009002, 76148146, 128687426

Offset: 0

Robert G. Wilson v, Jul 08 2004

nonn

Vladimir Pletser, Table of n, a(n) for n = 0..1000

Cf. A000045, A005181.

with (combinat): seq(round(ceil(exp((n/2)-1)))-fibonacci(n), n=0..50); # Vladimir Pletser Sep 15 2013

Table[ Ceiling[E^(n/2 - 1)] - Fibonacci[n], {n, 0, 41}]

A005181(n+1) - A000045(n).

A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 35, 58, 97, 163, 275, 466, 793, 1353, 2315, 3969, 6817, 11726, 20195, 34816, 60073, 103724, 179195, 309724, 535537, 926275, 1602515, 2773034, 4799353, 8307516, 14381675, 24899377, 43112257, 74651790, 129271235, 223862687, 387682633, 671402698, 1162785755, 2013837368, 3487832977, 6040770648, 10462450355, 18120829034, 31385253913, 54359521280, 94151567435, 163072632198

Offset: 0

Vladimir Pletser, Sep 15 2013

nonn

This sequence illustrates the second law of small numbers because it is a coincidence that the terms for 1 <= n <= 8 are the same as the Fibonacci numbers F(n) (A000045): a(n) = F(n) for 1 <= n <= 8.
Furthermore, the following terms are the sum of two Fibonacci numbers: a(9) = F(9) + F(2), a(10) = F(10) + F(4), a(11) = F(11) + F(6), a(14) = F(14) + F(11); or the algebraic sum of three Fibonacci numbers: a(12) = F(12) + F(8) - F(3), a(13) = F(13) + F(10) - F(7), a(14) = F(14) + F(12) - F(10), a(18) = F(19) - F(13) - F(8), a(19) = F(20) + F(10) - F(4); or the algebraic sum of four Fibonacci numbers: a(15) = F(15) + F(12) + F(9) + F(5), a(16) = F(16) + F(14) - F(6) - F(4), a(17) = F(18) - F(13) - F(9) - F(3), a(18) = F(18) + F(16) + F(14) + F(8), a(19) = F(19) + F(18) + F(10) - F(3).
Note that, for following values of n, a(n) > F(n+1) for n >= 20.
Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~=  Pi (see A135611).

T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001
I. Stewart, L'univers des nombres, Belin-Pour La Science, Paris 2000.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Cf. A000045, A005181.

[Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // Vincenzo Librandi, Sep 20 2013

seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);

Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* Vincenzo Librandi, Sep 20 2013 *)

a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).

A214076 a(n) = ceiling(e^(n/3)).

Original entry on oeis.org

2, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 77, 107, 149, 208, 290, 404, 564, 786, 1097, 1531, 2136, 2981, 4161, 5807, 8104, 11309, 15783, 22027, 30741, 42902, 59875, 83562, 116619, 162755, 227143, 317004, 442414, 617438

Offset: 1

Mohammad K. Azarian, Jul 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005181, A005182, A214077.

PROG(y := [], n := 60, LOOP(IF(n = 0, RETURN y), y := ADJOIN(CEILING(ê^(n/3)), y), n := n - 1))

[Ceiling(Exp(n)^(1/3)): n in [1..50]]; // Vincenzo Librandi, Feb 13 2013

A214076:=n->ceil(exp(n/3)): seq(A214076(n), n=1..60); # Wesley Ivan Hurt, Jan 11 2016

Ceiling[E^(Range[40]/3)] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = ceil(exp(n/3)); \\ Michel Marcus, Jan 11 2016

A214077 a(n) = floor(e^(n/3)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 39, 54, 76, 106, 148, 207, 289, 403, 563, 785, 1096, 1530, 2135, 2980, 4160, 5806, 8103, 11308, 15782, 22026, 30740, 42901, 59874, 83561, 116618, 162754, 227142, 317003, 442413, 617437

Offset: 1

Mohammad K. Azarian, Jul 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005181, A005182, A214076.

PROG(y := [], n := 60, LOOP(IF(n = 0, RETURN y), y := ADJOIN(FLOOR(ê^(n/3)), y), n := n - 1))

[Floor(Exp(n)^(1/3)): n in [1..50]]; // Vincenzo Librandi, Feb 13 2013

A214077:=n->floor(exp(n/3)): seq(A214077(n), n=1..60); # Wesley Ivan Hurt, Jan 11 2016

Floor[E^(Range[50]/3)] (* Vincenzo Librandi, Feb 13 2013 *)

a(n) = floor(exp(n/3)); \\ Michel Marcus, Jan 11 2016

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

Original entry on oeis.org

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

Alexei Kourbatov, Feb 18 2019

nonn

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.

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

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

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

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

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

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

cp's OEIS Frontend

A096766 Difference between ceiling(e^(n/2 - 1)) (A005181) and the n-th Fibonacci number (A000045).

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A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

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A214076 a(n) = ceiling(e^(n/3)).

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A214077 a(n) = floor(e^(n/3)).

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

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