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A264147 a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.

%I A264147 #41 Sep 08 2022 08:46:14
%S A264147 0,-1,1,1,5,10,22,43,83,155,285,516,924,1639,2885,5045,8773,15182,
%T A264147 26162,44915,76855,131119,223101,378696,641400,1084175,1829257,
%U A264147 3081193,5181893,8702290,14594830,24446971,40902299,68359619,114132765,190373580,317258388,528265207
%N A264147 a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.
%C A264147 a(n) is prime for n = 4, 7, 8, 26, 28, 52, 86, 87, 93, 97, 158, 196, 303, 2908, 3412, 4111, 4208, 6183, 6337, 9878, ...
%H A264147 Bruno Berselli, <a href="/A264147/b264147.txt">Table of n, a(n) for n = 0..500</a>
%H A264147 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A264147 G.f.:  x*(-1 + 3*x)/(1 - x - x^2)^2.
%F A264147 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
%F A264147 a(n) = n*F(n-1) - F(n).
%F A264147 a(n) = Sum_{i=0..n} F(i)*L(n-1-i), where L() is a Lucas number (A000032).
%F A264147 a(n) = 3*A001629(n) - A001629(n+1).
%F A264147 a(n) = -(-1)^n*A178521(-n).
%F A264147 a(n+2) - a(n) = A007502(n+1).
%F A264147 Sum_{i>0} 1/a(i) = 1.39516607051636028893879220294180374...
%F A264147 a(n) = (-((1+sqrt(5))/2)^n*(2*sqrt(5) + (-5+sqrt(5))*n) + ((1-sqrt(5))/2)^n*(2*sqrt(5) + (5+sqrt(5))*n)) / 10. - _Colin Barker_, Jul 27 2017
%F A264147 a(n) = (-i)^n*(n*sin(c*(n+1)) - (n+1)*sin(c*n)*i)/sqrt(5/4) where c = arccos(i/2). - _Peter Luschny_, May 16 2022
%p A264147 A264147 := proc(n)
%p A264147     n*combinat[fibonacci](n+1)-(n+1)*combinat[fibonacci](n) ;
%p A264147 end proc:
%p A264147 seq(A264147(n),n=0..10) ; # _R. J. Mathar_, Jun 02 2022
%t A264147 Table[n Fibonacci[n + 1] - (n + 1) Fibonacci[n], {n, 0, 40}]
%o A264147 (PARI) for(n=0, 40, print1(n*fibonacci(n+1)-(n+1)*fibonacci(n)", "));
%o A264147 (Sage) [n*fibonacci(n+1)-(n+1)*fibonacci(n) for n in (0..40)]
%o A264147 (Maxima) makelist(n*fib(n+1)-(n+1)*fib(n), n, 0, 40);
%o A264147 (Magma) [n*Fibonacci(n+1)-(n+1)*Fibonacci(n): n in [0..40]];
%o A264147 (PARI) concat(0, Vec(-x*(1 - 3*x) / (1 - x - x^2)^2 + O(x^50))) \\ _Colin Barker_, Jul 27 2017
%o A264147 (Julia) # The function 'fibrec' is defined in A354044.
%o A264147 function A264147(n)
%o A264147     n == 0 && return BigInt(0)
%o A264147     a, b = fibrec(n)
%o A264147     n*b - a*(n + 1)
%o A264147 end # _Peter Luschny_, May 16 2022
%Y A264147 Cf. A000045, A001629, A007502, A354044.
%Y A264147 Cf. A178521: n*F(n+1) + (n+1)*F(n).
%Y A264147 Cf. A094588: n*F(n-1) + F(n).
%Y A264147 Cf. A099920: Sum_{i=0..n} F(i)*L(n-i).
%Y A264147 Cf. A023607: Sum_{i=0..n} F(i)*L(n+1-i).
%K A264147 sign,easy
%O A264147 0,5
%A A264147 _Bruno Berselli_, Nov 04 2015