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A190173 a(n) = Sum_{1 <= i < j <= n} F(i)*F(j), where F(k) is the k-th Fibonacci number.

%I A190173 #32 Sep 08 2022 08:45:56
%S A190173 0,1,5,17,52,148,408,1101,2937,7777,20504,53912,141520,371113,972573,
%T A190173 2547825,6672876,17473996,45754280,119797205,313650865,821177281,
%U A190173 2149916400,5628629232,14736064032,38579712913,101003317493,264430632401,692289215332,1812438042052
%N A190173 a(n) = Sum_{1 <= i < j <= n} F(i)*F(j), where F(k) is the k-th Fibonacci number.
%H A190173 Vincenzo Librandi and Bruno Berselli, <a href="/A190173/b190173.txt">Table of n, a(n) for n = 1..1000</a> (First 211 terms from Vincenzo Librandi)
%H A190173 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-6,4,2,-1).
%F A190173 a(n) = F(n+1)^2 - F(n+2) + (1-(-1)^n)/2.
%F A190173 G.f.: x^2*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2)*(1-3*x+x^2)). - _Bruno Berselli_, Jun 20 2012
%e A190173 a(4) = F(1)*F(2) + F(1)*F(3) + F(1)*F(4) + F(2)*F(3) + F(2)*F(4) + F(3)*F(4) = 1 + 2 + 3 + 2 + 3 + 6 = 17.
%p A190173 with(combinat): seq(fibonacci(n+1)^2-fibonacci(n+2)+1/2-(1/2)*(-1)^n, n = 1 .. 30);
%t A190173 Table[Fibonacci[n + 1]^2 - Fibonacci[n + 1] + (1 - (-1)^n)/2, {n,1,50}] (* _G. C. Greubel_, Mar 04 2017 *)
%o A190173 (Magma) [Fibonacci(n+1)^2 - Fibonacci(n+2) + (1-(-1)^n)/2: n in [1..30]]; // _Vincenzo Librandi_, Jun 05 2011
%o A190173 (PARI) a(n)=fibonacci(n+1)^2-fibonacci(n+2)+n%2 \\ _Charles R Greathouse IV_, Jun 08 2011
%Y A190173 Cf. A000045.
%K A190173 nonn,easy
%O A190173 1,3
%A A190173 _Emeric Deutsch_, May 31 2011