A246173 -id:A246173 - cp's OEIS frontend

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

0, 1, 4, 16, 54, 176, 548, 1667, 4968, 14592, 42348, 121728, 347112, 983173, 2768812, 7758928, 21648546, 60172784, 166687436, 460356359, 1267964496, 3483818880, 9550754520, 26129950080, 71356349520, 194529354505, 529485228244, 1439096616976, 3906061968654, 10588691040176, 28670559059444

Offset: 0

Emeric Deutsch, Mar 26 2014

The Fibonacci cube G_n is defined in the Klavzar and Mollard reference (as Gamma_n).

			a(2)=4 because the Fibonacci cube G_2 is the path P_3 having Wiener index 1 + 1 + 2 = 4.

G. C. Greubel, Table of n, a(n) for n = 0..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, Fibonacci 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).

Cf. A000045.

[(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

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);

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 *)
LinearRecurrence[{4, 0, -10, 0, 4, -1}, {0, 1, 4, 16, 54, 176}, 100] (* G. C. Greubel, Mar 27 2016 *)
CoefficientList[Series[x/(1 - 2 x - 2 x^2 + x^3)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 07 2017 *)
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 *)

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

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).
a(n) = sum( F(i)*F(i+1)*F(n-i+1)*F(n-i+2), i=1..n ), where F = A000045.
Empirical g.f.: x / ((x+1)^2*(x^2-3*x+1)^2). - Colin Barker, Mar 26 2014
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.
The g.f. above is correct because it is equal to (dG/dt)A246173.%20-%20_Emeric%20Deutsch">{t=1}, where G is the bivariate g.f. given in A246173. - _Emeric Deutsch, Oct 01 2014
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
25*a(n) = n*Lucas(2*(n+2))+2*Fibonacci(2*n+1)-(n+2)*(-1)^n. - Ehren Metcalfe, Mar 26 2016
a(n) = 4*a(n-1) - 10*a(n-3) + 4*a(n-5) - a(n-6). - G. C. Greubel, Mar 27 2016
25*a(n) = -A001906(n+2) +5*A001871(n) -(n+2)*(-1)^n. - R. J. Mathar, Jul 24 2022

A246174 Triangle read by rows: T(n,k) is the number of vertex pairs at distance k of the Lucas cube Lambda(n) (1<=k<=n).

Original entry on oeis.org

2, 1, 3, 3, 8, 8, 4, 1, 15, 20, 15, 5, 30, 48, 44, 24, 6, 1, 56, 105, 119, 84, 35, 7, 104, 224, 296, 256, 144, 48, 8, 1, 189, 459, 696, 711, 495, 228, 63, 9, 340, 920, 1570, 1840, 1522, 880, 340, 80, 10, 1, 605, 1804, 3421, 4521, 4312, 2981, 1463, 484, 99, 11

Offset: 2

Emeric Deutsch, Aug 18 2014

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.
The number of entries in row n is equal to n if n is even and equal to n-1 if n is odd.
The entries in row n are the coefficients of the Hosoya polynomial of the Lucas cube Lambda(n).
T(n,1) = A099920(n-1) = number of edges in Lambda(n).
Sum(kT(n,k), k>=1) = A238420(n) = the Wiener index of Lambda(n).

			Row 2 is 2,1. Indeed, Lambda(2)  is the path-tree P(3) having vertex-pair distances 1,1, and 2.
Triangle starts:
2,1;
3,3;
8,8,4,1;
15,20,15,5;
30,48,44,24,6,1;

S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
E. Munarini, C. P. Cippo, N. Z. Salvi, On the Lucas cubes, The Fibonacci Quarterly, 39, No. 1, 2001, 12-21.

Cf. A099920, A238420, A246173.

g := t*z^2*(2+t-z+t*z-3*t*z^2+t*z^3+t*z^4)/((1+t*z)*(1-z-t*z-z^2-t*z^2+t*z^3)*(1-z-z^2)): gserz := simplify(series(g, z = 0, 20)): for j from 2 to 18 do H[j] := sort(coeff(gserz, z, j)) end do: for j from 2 to 13 do seq(coeff(H[j], t, k), k = 1 .. 2*floor((1/2)*j)) end do; # yields sequence in triangular form

G.f.: tz^2(2+t-z+tz-3tz^2+tz^3+tz^4)/((1+tz)(1-z-tz-z^2-tz^2+tz^3)(1-z-z^2)). Derived from Theorem 4.3 of the Klavzar-Mollard reference in which the g.f. of the ordered Hosoya polynomials is given.

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

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