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

Author

Amarnath Murthy, Aug 27 2005

Keywords

• nonn
• easy

Examples

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

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).

Crossrefs

Cf. A000045, A006002.

Programs

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 *)

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

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

Extensions

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