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

%I A109454 #30 Sep 30 2024 19:01:36
%S A109454 0,0,0,0,4,13,42,119,330,890,2376,6291,16588,43615,114492,300236,
%T A109454 786828,2061233,5398470,14136759,37015990,96917974,253748880,
%U A109454 664346375,1739318904,4553656703,11921726232,31211643384,81713400340,213928875445,560073740226
%N A109454 Sum of non-Fibonacci numbers between successive Fibonacci numbers: a(n) = Sum_{k=F(n)+1..F(n+1)-1} k.
%H A109454 Colin Barker, <a href="/A109454/b109454.txt">Table of n, a(n) for n = 0..1000</a>
%H A109454 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H A109454 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).
%F A109454 a(n) = Fibonacci(n+2)*(Fibonacci(n-1)-1)/2, n>1. - _Vladeta Jovovic_, Aug 27 2005
%F A109454 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
%F A109454 G.f.: x^4*(x^2-x-4) / ((x+1)*(x^2-3*x+1)*(x^2+x-1)). - _Colin Barker_, Mar 26 2015
%e A109454 F(5) = F(4) + 1 = 4.
%e A109454 F(6) = (F(5) + 1) + (F(5) + 2) = 6+7 = 13.
%e A109454 F(7) = 9+10+11+12 = 42.
%t A109454 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 *)
%t A109454 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 *)
%o A109454 (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
%Y A109454 Cf. A000045, A006002.
%K A109454 nonn,easy
%O A109454 0,5
%A A109454 _Amarnath Murthy_, Aug 27 2005
%E A109454 More terms from _Franklin T. Adams-Watters_, Jun 06 2006