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A238419 a(n) = the Wiener index of the Fibonacci cube G_n.

%I A238419 #58 Feb 16 2025 08:33:21
%S A238419 0,1,4,16,54,176,548,1667,4968,14592,42348,121728,347112,983173,
%T A238419 2768812,7758928,21648546,60172784,166687436,460356359,1267964496,
%U A238419 3483818880,9550754520,26129950080,71356349520,194529354505,529485228244,1439096616976,3906061968654,10588691040176,28670559059444
%N A238419 a(n) = the Wiener index of the Fibonacci cube G_n.
%C A238419 The Fibonacci cube G_n is defined in the Klavzar and Mollard reference (as Gamma_n).
%H A238419 G. C. Greubel, <a href="/A238419/b238419.txt">Table of n, a(n) for n = 0..1000</a>
%H A238419 S. Klavzar, M. Mollard, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match68/n1/match68n1_311-324.pdf">Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes</a>, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
%H A238419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciCubeGraph.html">Fibonacci Cube Graph</a>
%H A238419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H A238419 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-10,0,4,-1).
%F A238419 a(n) = ( 4*(n + 1)*F(n)^2 + (9*n + 2)*F(n)*F(n+1) + 6*n*F(n+1)^2 )/25, where F = A000045 (Fibonacci numbers).
%F A238419 a(n) = sum( F(i)*F(i+1)*F(n-i+1)*F(n-i+2), i=1..n ), where F = A000045.
%F A238419 Empirical g.f.: x / ((x+1)^2*(x^2-3*x+1)^2). - _Colin Barker_, Mar 26 2014
%F A238419 The g.f. above is correct because a(n) is the convolution of F(n)F(n+1) by F(n+1)F(n+2) (see Klavzar and Mollard paper). - _Michel Mollard_, Aug 20 2014.
%F A238419 The g.f. above is correct because it is equal to (dG/dt)_{t=1}, where G is the bivariate g.f. given in A246173. - _Emeric Deutsch_, Oct 01 2014
%F A238419 a(n+3) = (2*(n+3)*a(n+2)+2*(n+4)*a(n+1)-(n+5)*a(n))/(n+2). - _Robert Israel_, Aug 29 2014
%F A238419 25*a(n) = n*Lucas(2*(n+2))+2*Fibonacci(2*n+1)-(n+2)*(-1)^n. - _Ehren Metcalfe_, Mar 26 2016
%F A238419 a(n) = 4*a(n-1) - 10*a(n-3) + 4*a(n-5) - a(n-6). - _G. C. Greubel_, Mar 27 2016
%F A238419 25*a(n) = -A001906(n+2) +5*A001871(n) -(n+2)*(-1)^n. - _R. J. Mathar_, Jul 24 2022
%e A238419 a(2)=4 because the Fibonacci cube G_2 is the path P_3 having Wiener index 1 + 1 + 2 = 4.
%p A238419 with(combinat): F := proc (n) options operator, arrow: Fibonacci(n) end proc: a := proc (n) options operator, arrow: (1/25)*(4*n+4)*F(n)^2+(1/25)*(9*n+2)*F(n)*F(n+1)+(6/25)*n*F(n+1)^2 end proc: seq(a(n), n = 0 .. 30);
%t A238419 wifc[n_]:=Module[{f1=Fibonacci[n],f2=Fibonacci[n+1]},(4*(n+1)*f1^2+ (9*n+2)*f1*f2+ 6*n*f2^2)/25]; wifc[Range[0, 30]] (* _Harvey P. Dale_, Sep 22 2014 *)
%t A238419 LinearRecurrence[{4, 0, -10, 0, 4, -1}, {0, 1, 4, 16, 54, 176}, 100] (* _G. C. Greubel_, Mar 27 2016 *)
%t A238419 CoefficientList[Series[x/(1 - 2 x - 2 x^2 + x^3)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 07 2017 *)
%t A238419 Table[(n LucasL[2 (n + 2)] + 2 Fibonacci[2 n + 1] - (n + 2) (-1)^n)/25, {n, 0, 20}] (* _Eric W. Weisstein_, Sep 07 2017 *)
%o A238419 (Magma) [(4*(n+1)*Fibonacci(n)^2 + (9*n+2)*Fibonacci(n)*Fibonacci(n+1) + 6*n*Fibonacci(n+1)^2)/25: n in [0..30]]; // _Vincenzo Librandi_, Mar 27 2014
%o A238419 (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,4,0,-10,0,4]^n*[0;1;4;16;54;176])[1,1] \\ _Charles R Greathouse IV_, Sep 07 2017
%Y A238419 Cf. A000045.
%K A238419 nonn,easy
%O A238419 0,3
%A A238419 _Emeric Deutsch_, Mar 26 2014