A004696 -id:A004696 - cp's OEIS frontend

A293543 a(n) = ceiling(Fibonacci(n)/3).

0, 1, 1, 1, 1, 2, 3, 5, 7, 12, 19, 30, 48, 78, 126, 204, 329, 533, 862, 1394, 2255, 3649, 5904, 9553, 15456, 25009, 40465, 65473, 105937, 171410, 277347, 448757, 726103, 1174860, 1900963, 3075822, 4976784, 8052606, 13029390, 21081996, 34111385, 55193381

Offset: 0

Clark Kimberling, Oct 12 2017

a(n) is the least integer k such that k/Fibonacci(n) > 1/3.

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

Cf. A000045.
Cf. A004696 (floor(Fibonacci(n)/3)).
Cf. A293544 (round(Fibonacci(n)/3)).

LinearRecurrence[{1, 1, 0, 0, 0, 0, 0, 1, -1, -1}, {0, 1, 1, 1, 1, 2, 3, 5, 7, 12}, 50] (* Harvey P. Dale, Oct 18 2018 *)
Table[Ceiling[Fibonacci[n]/3], {n, 0, 20}] (* Eric W. Weisstein, Feb 07 2025 *)
Ceiling[Fibonacci[Range[0, 20]]/3] (* Eric W. Weisstein, Feb 07 2025 *)
CoefficientList[Series[-x (-1 + x^2 + x^3 + x^7 + x^8)/((-1 + x) (1 + x) (1 + x^2) (-1 + x + x^2) (1 + x^4)), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 07 2025 *)
Table[(9 - 6 Cos[n Pi/2] + 8 Fibonacci[n] - (-1)^n (3 + 4 Sin[n Pi/4] (Cos[n Pi/2] + Sqrt[2] Sin[n Pi/2])))/24, {n, 0, 20}] (* Eric W. Weisstein, Feb 07 2025 *)

G.f.: -((x (-1 + x^2 + x^3 + x^7 + x^8))/((-1 + x) (1 + x) (1 + x^2) (-1 + x + x^2) (1 + x^4))).

a(n) = a(n-1) + a(n-2) + a(n-8) - a(n-9) - a(n-10) for n >= 11.

A293544 a(n) = round(Fibonacci(n)/3).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 7, 11, 18, 30, 48, 78, 126, 203, 329, 532, 861, 1394, 2255, 3649, 5904, 9552, 15456, 25008, 40464, 65473, 105937, 171410, 277347, 448756, 726103, 1174859, 1900962, 3075822, 4976784, 8052606, 13029390, 21081995, 34111385, 55193380

Offset: 0

Clark Kimberling, Oct 12 2017

a(n) is the integer k that minimizes | k/Fibonacci(n) - 1/3 |.

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

Cf. A000045 (Fibonacci(n)).
Cf. A004696 (floor(Fibonacci(n)/3)).
Cf. A293543 (ceiling(Fibonacci(n)/3)).

Table[Round[Fibonacci[n]/3], {n, 0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)
Round[Fibonacci[Range[0, 20]]/3] (* Eric W. Weisstein, Feb 08 2025 *)
LinearRecurrence[{1, 1, 0, -1, 1, 1}, {0, 0, 1, 1, 2, 3}, {0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)
CoefficientList[Series[-(x^3/((-1 + x + x^2) (1 + x^4))), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 08 2025 *)
Table[(Fibonacci[n] + (-1)^n Sin[n Pi/4] (Cos[n Pi/2] + Sqrt[2] Sin[n Pi/2]))/3, {n, 0, 20}] (* Eric W. Weisstein, Feb 08 2025 *)

G.f.: -(x^2/((-1 + x + x^2) (1 + x^4))).

a(n) = a(n-1) + a(n-2) - a(n-4) + a(n-5) + a(n-6) for n >= 7.

A179001 Partial sums of floor(Fibonacci(n)/3).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 8, 15, 26, 44, 73, 121, 198, 323, 526, 855, 1387, 2248, 3641, 5896, 9544, 15447, 24999, 40455, 65463, 105927, 171399, 277336, 448745, 726091, 1174847, 1900950, 3075809, 4976771, 8052592, 13029376, 21081981, 34111370, 55193365, 89304750

Offset: 0

Mircea Merca, Jan 03 2011

nonn

Partial sums of A004696.

			a(9) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 4 + 7 + 11 = 26.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

Cf. A004696.

[Floor(Fibonacci(n+2)/3-3*n/8-1/6): n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

A179001 := proc(n) add( floor(combinat[fibonacci](i)/3),i=0..n) ; end proc:

Accumulate[Floor[Fibonacci[Range[0,40]]/3]] (* Harvey P. Dale, Jun 13 2022 *)

a(n) = round(Fibonacci(n+2)/3 - 3*n/8 - 11/24).
a(n) = round(Fibonacci(n+2)/3 - 3*n/8 - 1/3).
a(n) = floor(Fibonacci(n+2)/3 - 3*n/8 - 1/6).
a(n) = ceiling(Fibonacci(n+2)/3 - 3*n/8 - 3/4).
a(n) = a(n-8) + Fibonacci(n-1) + Fibonacci(n-3) - 3, n > 8.
a(n) = 2*a(n-1) - a(n-3) + a(n-8) - 2*a(n-9) + a(n-11), n > 10.
G.f.: -x^4*(1 + x^4 + x^3) / ( (1+x)*(x^2+1)*(x^2+x-1)*(x^4+1)*(x-1)^2 ).

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A293543 a(n) = ceiling(Fibonacci(n)/3).

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A293544 a(n) = round(Fibonacci(n)/3).

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A179001 Partial sums of floor(Fibonacci(n)/3).

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