A109454 Sum of non-Fibonacci numbers between successive Fibonacci numbers: a(n) = Sum_{k=F(n)+1..F(n+1)-1} k.

0, 0, 0, 0, 4, 13, 42, 119, 330, 890, 2376, 6291, 16588, 43615, 114492, 300236, 786828, 2061233, 5398470, 14136759, 37015990, 96917974, 253748880, 664346375, 1739318904, 4553656703, 11921726232, 31211643384, 81713400340, 213928875445, 560073740226

Offset

0,5

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Formula

a(n) = Fibonacci(n+2)*(Fibonacci(n-1)-1)/2, n>1. - Vladeta Jovovic, Aug 27 2005

a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>6. - Colin Barker, Mar 26 2015

G.f.: x^4*(x^2-x-4) / ((x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Mar 26 2015

Example

F(5) = F(4) + 1 = 4.
F(6) = (F(5) + 1) + (F(5) + 2) = 6+7 = 13.
F(7) = 9+10+11+12 = 42.

Mathematica

CoefficientList[Series[x^4*(x^2 - x - 4)/((x + 1) (x^2 - 3 x + 1) (x^2 + x - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 08 2021 *)
Total[Range[#[[1]]+1, #[[2]]-1]]&/@Partition[Fibonacci[Range[0, 40]], 2, 1] (* or *) LinearRecurrence[{3, 1, -5, -1, 1}, {0, 0, 0, 0, 4, 13, 42}, 40] (* Harvey P. Dale, Sep 30 2024 *)

Prog

(PARI) concat([0, 0, 0, 0], Vec(x^4*(x^2-x-4) / ((x+1)*(x^2-3*x+1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Mar 26 2015

Crossrefs

Cf. A000045, A006002.

Sequence in context: A036366 A303863 A255836 * A357063 A307261 A287349

Adjacent sequences: A109451 A109452 A109453 * A109455 A109456 A109457

Keyword

nonn,easy

Author

Amarnath Murthy, Aug 27 2005

Extensions

More terms from Franklin T. Adams-Watters, Jun 06 2006

Status

approved