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%I #47 Mar 31 2022 03:13:27
%S 0,1,7,31,115,390,1254,3893,11789,35045,102695,297516,853932,2432041,
%T 6881395,19361995,54214939,151164018,419910354,1162585565,3209268665,
%U 8835468881,24266461007,66501634776,181882282200,496539007825,1353272290399,3682496714743
%N a(n) = Sum_{k=0..n} k*Fibonacci(2*k).
%C There are only a small number of Fibonacci identities that can be solved for n. Some of these are
%C 1. n = (-F(4*n) + 5*Sum_{k=1..n} F(2*k-1)^2)/2 (Vajda #95).
%C 2. n = (F(n+3) - 2 + Sum_{k=0..n} k*F(k))/F(n+2). (A104286)
%C 3. n = (a(n) + F(2*n))/F(2*n+1).
%C 4. n = F(n+4) - 3 - Sum_{k=0..1} (F(k+2) - 1). (A001924)
%C n can also be expressed in terms of phi=(1+sqrt(5))/2:
%C 5. n = floor(n*phi^3) - floor(2*n*phi).
%C 6. n = (floor(2*n*phi^2) - floor(2*n*phi))/2.
%H Michael De Vlieger, <a href="/A197649/b197649.txt">Table of n, a(n) for n = 0..2384</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2011.10827">Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers</a>, arXiv:2011.10827 [math.CO], 2020.
%H Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/2203.13205">Honeycombs in the Pascal triangle and beyond</a>, arXiv:2203.13205 [math.HO], 2022. See p. 5.
%H E. Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com.es/2010/05/small-fibonacci-sum_13.html">A small Fibonacci sum</a>, Psychedelic Geometry Blogspot
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F a(n) = n*F(2*n+1) - F(2*n), where F(n)= Fibonacci(n).
%F a(n) = ((F(2*n+1)*((n-1)*h(n-1) - (n-1)*h(n-2)) - h(n)*F(2*n))/h(n), n > 2, where h(n) is the n-th harmonic number.
%F From _R. J. Mathar_, Oct 17 2011: (Start)
%F G.f.: x*(1+x) / ( (x^2-3*x+1)^2 ).
%F a(n) = A001871(n-1) + A001871(n-2). (End)
%F a(n) ~ c*n*(3 + sqrt(5))^n*2^(-n), where c = (5 + sqrt(5))/10. - _Stefano Spezia_, Mar 29 2022
%p a:=n->sum(k*fibonacci(2*k),n= 0..n):seq(a(n), n=0..25);
%t Table[Sum[k*Fibonacci[2*k], {k, 0, n}], {n, 0, 50}] (* _T. D. Noe_, Oct 17 2011 *)
%Y Cf. A023619 (inverse binomial transform).
%Y Cf. A001871, A001924, A104286.
%K nonn,easy
%O 0,3
%A _Gary Detlefs_, Oct 16 2011
%E Identity 4 added by _Gary Detlefs_, Dec 22 2012
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