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A354044 a(n) = 2*(-i)^n*(n*sin(c*(n+1)) - i*sin(-c*n))/sqrt(5) where c = arccos(i/2).

%I A354044 #30 May 18 2022 10:28:03
%S A354044 0,2,5,11,23,45,86,160,293,529,945,1673,2940,5134,8917,15415,26539,
%T A354044 45525,77842,132716,225685,382877,648165,1095121,1846968,3109850,
%U A354044 5228261,8777315,14716223,24643389,41220110,68873848,114964805,191719849,319436697,531789785
%N A354044 a(n) = 2*(-i)^n*(n*sin(c*(n+1)) - i*sin(-c*n))/sqrt(5) where c = arccos(i/2).
%H A354044 Jianing Song, <a href="/A354044/b354044.txt">Table of n, a(n) for n = 0..1000</a>
%H A354044 Peter Luschny, <a href="/A354044/a354044.jpg">Illustration of A354044</a>.
%H A354044 Peter Luschny, <a href="https://www.luschny.de/math/seq/oeis/FibonacciFunction.html">The Fibonacci Function</a>.
%H A354044 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A354044 a(n) = [x^n] ((2 - x)*x*(x + 1))/(x^2 + x - 1)^2.
%F A354044 a(n) = (((-1 - sqrt(5))^(-n)*(sqrt(5)*n - n - 2) + (-1 + sqrt(5))^(-n)*(sqrt(5)*n + n + 2)))/(2^(1 - n)*sqrt(5)).
%F A354044 a(n) = (-1)^(n - 1)*(Fibonacci(-n) - n*Fibonacci(-n - 1)).
%F A354044 a(n) = (-1)^(n - 1)*A353595(-n, -n). (A353595 is defined for all n in Z.)
%F A354044 a(n) = ((-42*n^2 + 259*n - 350)*a(n - 3) + (123*n^2 - 76*n - 446)*a(n - 2) + (207*n^2 - 885*n + 412)*a(n - 1)) / ((165*n - 542)*(n - 1)) for n >= 4.
%F A354044 a(n) = Fibonacci(n) + n*Fibonacci(n+1). - _Jianing Song_, May 16 2022
%p A354044 c := arccos(I/2): a := n -> 2*(-I)^n*(n*sin(c*(n+1)) - I*sin(-c*n))/sqrt(5):
%p A354044 seq(simplify(a(n)), n = 0..35);
%o A354044 (Julia)
%o A354044 function fibrec(n::Int)
%o A354044     n == 0 && return (BigInt(0), BigInt(1))
%o A354044     a, b = fibrec(div(n, 2))
%o A354044     c = a * (b * 2 - a)
%o A354044     d = a * a + b * b
%o A354044     iseven(n) ? (c, d) : (d, c + d)
%o A354044 end
%o A354044 function A354044(n)
%o A354044     n == 0 && return BigInt(0)
%o A354044     a, b = fibrec(n + 1)
%o A354044     a*(n - 1) + b
%o A354044 end
%o A354044 println([A354044(n) for n in 0:35])
%o A354044 (PARI) a(n) = fibonacci(n) + n*fibonacci(n+1) \\ _Jianing Song_, May 16 2022
%Y A354044 Cf. A000045 (the Fibonacci numbers), A007502, A088209, A094588, A136391, A178521, A264147, A353595.
%K A354044 nonn,easy
%O A354044 0,2
%A A354044 _Peter Luschny_, May 16 2022