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

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

Author

Clark Kimberling, Oct 12 2017

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A000045 (Fibonacci(n)).

Cf. A004696 (floor(Fibonacci(n)/3)).

Cf. A293543 (ceiling(Fibonacci(n)/3)).

Programs

Mathematica

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 *)

Formula

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.