A094707 Partial sums of repeated Fibonacci sequence.

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

Offset: 0

Author

Paul Barry, May 21 2004

Keywords

• easy
• nonn

Comments

Equals row sums of triangle A139147 starting with "1". - Gary W. Adamson, Apr 11 2008

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).

Crossrefs

Cf. A000045, A103609, A131524, A139147.

Programs

Magma

[Fibonacci(Floor((n+6)/2))*((n+1) mod 2) + 2*Fibonacci(Floor((n+3)/2))*(n mod 2) - 2: n in [0..60]]; // G. C. Greubel, Feb 12 2023

Mathematica

LinearRecurrence[{1,1,-1,1,-1}, {0,0,1,2,3}, 50] (* Jean-François Alcover, Nov 18 2017 *)

SageMath

def A094707(n): return fibonacci((n+6)//2) - 2 if (n%2==0) else 2*fibonacci((n+3)//2) - 2
[A094707(n) for n in range(61)] # G. C. Greubel, Feb 12 2023

Formula

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

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

a(n) = Sum_{k=0..n} Fibonacci(floor(k/2)).

a(n) = -2 - (sqrt(5)/2 - 1/2)^(n/2)*((2*sqrt(5)/5 - 1)*cos(Pi*n/2) + sqrt(4*sqrt(5)/5 - 8/5)*sin(Pi*n/2)) - (sqrt(5)/2 + 1/2)^(n/2)*((sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5 - 1/2)*(-1)^n - sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5-1/2).

a(n) = A131524(n) + A131524(n+1). - R. J. Mathar, Jul 07 2011

a(n) = Fibonacci(n/2 +3) - 2 if n even, otherwise a(n) = 2*Fibonacci((n-1)/2 + 2) - 2. - G. C. Greubel, Feb 12 2023