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A023552 Convolution of natural numbers >= 3 and Fibonacci numbers.

%I A023552 #29 Nov 25 2024 17:56:52
%S A023552 3,7,15,28,50,86,145,241,397,650,1060,1724,2799,4539,7355,11912,19286,
%T A023552 31218,50525,81765,132313,214102,346440,560568,907035,1467631,2374695,
%U A023552 3842356,6217082,10059470,16276585,26336089,42612709,68948834,111561580,180510452
%N A023552 Convolution of natural numbers >= 3 and Fibonacci numbers.
%H A023552 Colin Barker, <a href="/A023552/b023552.txt">Table of n, a(n) for n = 1..1000</a>
%H A023552 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A023552 G.f.: x*(3-2*x)/((1-x-x^2)*(1-x)^2). - _Ralf Stephan_, Apr 28 2004
%F A023552 From _Colin Barker_, Mar 11 2017: (Start)
%F A023552 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
%F A023552 a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-13+5*sqrt(5)) + (1+sqrt(5))^n*(13+5*sqrt(5)))) / sqrt(5) - n. (End)
%F A023552 a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n+5). - _G. C. Greubel_, Jul 08 2019
%t A023552 LinearRecurrence[{3,-2,-1,1},{3,7,15,28},40] (* or *) Rest[ CoefficientList[Series[(x(3-2x))/((1-x-x^2)(1-x)^2),{x,0,40}],x]]  (* _Harvey P. Dale_, Apr 24 2011 *)
%t A023552 With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-n-5, {n,40}]] (* _G. C. Greubel_, Jul 08 2019 *)
%o A023552 (PARI) Vec(x*(3-2*x)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ _Colin Barker_, Mar 11 2017
%o A023552 (PARI) vector(40, n, f=fibonacci; f(n+4)+2*f(n+2)-n-5) \\ _G. C. Greubel_, Jul 08 2019
%o A023552 (Magma) F:=Fibonacci; [F(n+4)+2*F(n+2)-n-5: n in [1..40]]; // _G. C. Greubel_, Jul 08 2019
%o A023552 (Sage) f=fibonacci; [f(n+4)+2*f(n+2)-n-5 for n in (1..40)] # _G. C. Greubel_, Jul 08 2019
%o A023552 (GAP) F:=Fibonacci; List([1..40], n-> F(n+4)+2*F(n+2)-n-5); # _G. C. Greubel_, Jul 08 2019
%Y A023552 Cf. A000032, A000045, A213579.
%K A023552 nonn,easy
%O A023552 1,1
%A A023552 _Clark Kimberling_