A004697 -id:A004697 - cp's OEIS frontend

A293552 a(n) is the least integer k such that k/Fibonacci(n) > 1/4.

0, 1, 1, 1, 1, 2, 2, 4, 6, 9, 14, 23, 36, 59, 95, 153, 247, 400, 646, 1046, 1692, 2737, 4428, 7165, 11592, 18757, 30349, 49105, 79453, 128558, 208010, 336568, 544578, 881145, 1425722, 2306867, 3732588, 6039455, 9772043, 15811497, 25583539, 41395036, 66978574

Offset: 0

Clark Kimberling, Oct 14 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 1, -1, -1)

Cf. A000045, A293552, A293553.

z = 120; r = 1/4; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004697 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293552 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293553 *)

G.f.: -((x (-1 + x^2 + x^3 + x^5 + x^6))/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2))).
a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.
a(n) = ceiling(Fibonacci(n)/4).
a(n) = A004697(n) + 1 for n > 0.

A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 14, 22, 36, 58, 94, 152, 247, 399, 646, 1045, 1691, 2736, 4428, 7164, 11592, 18756, 30348, 49104, 79453, 128557, 208010, 336567, 544577, 881144, 1425722, 2306866, 3732588, 6039454, 9772042, 15811496, 25583539, 41395035, 66978574

Offset: 0

Clark Kimberling, Oct 14 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 1, -1, -1)

Cf. A000045, A004697, A131132, A293552.

z = 120; r = 1/4; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004697 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293552 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293553 *)

G.f.: x^4/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2)).
a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.
a(n) = floor(1/2 + Fibonacci(n)/4).
a(n) = A004697(n) if (fractional part of Fibonacci(n)/4) < 1/2, otherwise a(n) = A293552(n).
a(n) = A131132(n-4) for n > 3. - Georg Fischer, Oct 22 2018

A179006 Partial sums of floor(Fibonacci(n)/4).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 6, 11, 19, 32, 54, 90, 148, 242, 394, 640, 1039, 1685, 2730, 4421, 7157, 11584, 18748, 30340, 49096, 79444, 128548, 208000, 336557, 544567, 881134, 1425711, 2306855, 3732576, 6039442, 9772030, 15811484

Offset: 0

Mircea Merca, Jan 03 2011

nonn

Partial sums of A004697.

			a(7) = 0 + 0 + 0 + 0 + 0 + 1 + 2 + 3 = 6.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,2,-1,-1,1).

Cf. A004697.

A179006 := proc(n) add( floor(combinat[fibonacci](i)/4),i=0..n) ; end proc:

f[n_] := Floor[ Fibonacci@ n/4]; Accumulate@ Array[f, 38]
LinearRecurrence[{3,-3,2,-2,2,-1,-1,1},{0,0,0,0,0,1,3,6},40] (* Harvey P. Dale, Jan 28 2020 *)

a(n)={round(fibonacci(n+2)/4 - n/3 - 3/8)} \\ Andrew Howroyd, May 01 2020

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) - a(n-7) + a(n-8).
a(n) = round(Fibonacci(n+2)/4 - n/3 - 3/8).
a(n) = round(Fibonacci(n+2)/4 - n/3 - 1/4).
a(n) = floor(Fibonacci(n+2)/4 - n/3 - 1/12).
a(n) = ceiling(Fibonacci(n+2)/4 - n/3 - 2/3).
a(n) = a(n-6) + Fibonacci(n-1) - 2, n > 6.
G.f.: -x^5/((x^2+x+1)*(x^2-x+1)*(x^2+x-1)*(x-1)^2).

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A293552 a(n) is the least integer k such that k/Fibonacci(n) > 1/4.

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A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.

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A179006 Partial sums of floor(Fibonacci(n)/4).

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