A238420 a(n) is the Wiener index of the Lucas cube L_n.

0, 4, 9, 40, 120, 390, 1176, 3536, 10395, 30260, 87120, 248844, 705744, 1989820, 5581485, 15586720, 43356936, 120187026, 332134440, 915304520, 2516113215, 6900949484, 18888143904, 51599794200, 140718765600, 383142771700, 1041660829521, 2828107288216, 7668512468760, 20768716848030, 56185646831160, 151840963183424, 409947452576739, 1105779284582180, 2980113861417840

Offset: 1

Author

Emeric Deutsch, Mar 26 2014

Keywords

• nonn
• easy

Comments

The Lucas cube L_n is defined in the Klavzar and Mollard reference (as Lambda_n).

Examples

			a(2)=4 because the Lucas cube L_2 is the path P_3 having Wiener index 1 + 1 + 2 = 4.
a(3)=9 because the Lucas cube L_3 is the star on 4 vertices having Wiener index 1 + 1 + 1 + 2 + 2 + 2 = 9.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
• Eric Weisstein's World of Mathematics, Lucas Cube Graph
• Eric Weisstein's World of Mathematics, Wiener Index
• Index entries for linear recurrences with constant coefficients, signature (4,0,-10,0,4,-1).

Crossrefs

Cf. A000045.

Programs

Magma

[n*Fibonacci(n-1)*Fibonacci(n+1): n in [1..40]]; // Vincenzo Librandi, Mar 30 2014

Mathematica

Table[n Fibonacci[n - 1] Fibonacci[n + 1], {n, 1, 40}] (* Vincenzo Librandi, Mar 30 2014 *)
Table[(3 (-1)^n + LucasL[2 n]) n/5, {n, 15}] (* Eric W. Weisstein, Jul 29 2023 *)
LinearRecurrence[{4, 0, -10, 0, 4, -1}, {0, 4, 9, 40, 120, 390}, 20] (* Eric W. Weisstein, Jul 29 2023 *)
CoefficientList[Series[x (4 - 7 x + 4 x^2)/(1 - 2 x - 2 x^2 + x^3)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 29 2023 *)

Formula

a(n) = n * F(n-1) * F(n+1) where F(n)=A000045(n) are the Fibonacci numbers.

a(n) = (1/5) * ((4n+4)*F(2n-2) + (7n+7)*F(2n-1) - 3(n+1)*(-1)^n). - Ralf Stephan, Mar 30 2014

G.f.: (4*x^3 - 7*x^2 + 4*x)/((x + 1)^2 * (x^2 - 3*x + 1)^2). - Ralf Stephan, Mar 30 2014