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

Author

Clark Kimberling, Oct 12 2017

Keywords

• nonn
• easy

Comments

a(n) is the least integer k such that 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 (1,1,0,0,0,0,0,1,-1,-1)

Crossrefs

Cf. A000045.

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

Cf. A293544 (round(Fibonacci(n)/3)).

Programs

Mathematica

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 *)

Formula

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.