A355019 Partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + ..., where L = A000032 and F = A000045.

1, 0, 3, 2, 6, 4, 11, 8, 19, 14, 32, 24, 53, 40, 87, 66, 142, 108, 231, 176, 375, 286, 608, 464, 985, 752, 1595, 1218, 2582, 1972, 4179, 3192, 6763, 5166, 10944, 8360, 17709, 13528, 28655, 21890, 46366, 35420, 75023, 57312, 121391, 92734, 196416, 150048

Offset

0,3

Comments

The closely related partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ... are given by A355018.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Formula

a(n) = -2 + 2 F((n+3)/2) if n is odd, a(n) = - 2 + F((n+8)/2) if n is even, where F = A000045 (Fibonacci numbers).

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.

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

From G. C. Greubel, Mar 17 2024: (Start)

a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Lucas(floor(j/2) +1) - (1-(-1)^j) *Fibonacci(floor((j+1)/2)) ).

a(n) = (1/2)*( (1+(-1)^n)*Fibonacci(floor(n/2) +4) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ) - 2. (End)

Example

a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 3 = 3
a(3) = 1 - 1 + 3  - 1 = 2.

Mathematica

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* A355018 *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* this sequence *)
LinearRecurrence[{1, 1, -1, 1, -1}, {1, 0, 3, 2, 6}, 61] (* G. C. Greubel, Mar 17 2024 *)

Prog

(Magma) F:=Fibonacci; [(((n+1) mod 2)*F(Floor(n/2)+4) + 2*(n mod 2)*F(Floor((n+3)/2))) - 2: n in [0..60]]; // G. C. Greubel, Mar 17 2024
(SageMath) f=fibonacci; [(((n+1)%2)*f((n//2)+4) +2*(n%2)*f((n+3)//2)) -2 for n in range(61)] # G. C. Greubel, Mar 17 2024

Crossrefs

Cf. A000032, A000045, A355018, A355020, A355021.

Sequence in context: A160795 A258212 A347704 * A092401 A222208 A302853

Adjacent sequences: A355016 A355017 A355018 * A355020 A355021 A355022

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2022

Status

approved