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%I #36 Sep 08 2022 08:45:56
%S 0,1,1,4,8,18,36,71,135,252,462,836,1496,2653,4669,8164,14196,24566,
%T 42332,72675,124355,212156,360986,612744,1037808,1754233,2959801,
%U 4985476,8384480,14080602,23614932,39556031,66181311,110608188,184670694
%N a(n) = n*Fibonacci(n) - Sum_{i=0..n-1} Fibonacci(i).
%H Bruno Berselli, <a href="/A190062/b190062.txt">Table of n, a(n) for n = 0..1000</a>
%H Carlos Alirio Rico Acevedo, Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F G.f.: x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2).
%F a(n) = A045925(n) - A000071(n+1).
%F a(n) = (n-1)*Fibonacci(n) - Fibonacci(n-1) + 1.
%F a(n) = (((2*n-1)*r-5)*(1+r)^n-((2*n-1)*r+5)*(1-r)^n)/(10*2^n)+1, where r=sqrt(5).
%t CoefficientList[Series[x (1 - 2 x + 2 x^2) / ((1 - x) (1 - x - x^2)^2), {x, 0, 35}], x] (* _Vincenzo Librandi_, Aug 19 2013 *)
%o (Magma) [0] cat [n*Fibonacci(n)-(&+[Fibonacci(k): k in [0..n-1]]): n in [1..34]];
%o (PARI) concat(0, Vec(x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ _Altug Alkan_, Nov 13 2015
%Y Cf. A122491, A045925, A000071.
%K nonn,easy
%O 0,4
%A _Bruno Berselli_, May 04 2011
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