A245960 -id:A245960 - cp's OEIS frontend

A258121 Number of vertices of degree n in all Lucas cubes.

2, 5, 15, 39, 102, 267, 699, 1830, 4791, 12543, 32838, 85971, 225075, 589254, 1542687, 4038807, 10573734, 27682395, 72473451, 189737958, 496740423, 1300483311, 3404709510, 8913645219, 23336226147, 61095033222, 159948873519, 418751587335, 1096305888486, 2870166078123

Offset: 0

Emeric Deutsch, Jun 23 2015

Column sums of A245960.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sandi Klavzar, Michel Mollard and Marko Petkovsek, The degree sequence of Fibonacci and Lucas cubes, Discrete Math., Vol. 311, No. 14 (2011), pp. 1310-1322.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045, A001519, A022086, A245960.

I:=[2,5,15,39]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 19 2017

g := (2-x)*(1+x^2)/(1-3*x+x^2): gser := series(g, x = 0, 35): seq(coeff(gser, x, n), n = 0 .. 32);
with(combinat): 2, 5, seq(3*fibonacci(2*n+1), n = 2 .. 32);

CoefficientList[Series[(2 - x)*(1 + x^2)/(1 - 3 x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 19 2017 *)
Join[{2, 5}, LinearRecurrence[{3, -1}, {15, 39}, 30]] (* Vincenzo Librandi, Oct 19 2017 *)

my(x='x+O('x^50)); Vec((2-x)*(1+x^2)/(1-3*x+x^2)) \\ G. C. Greubel, Oct 19 2017

G.f.: (2-x)*(1+x^2)/(1-3*x+x^2).
a(n) = 3*F(2n+1) = 3*A001519(n+1) = A022086(2n+1) for n>=2; F(n) = A000045(n) are the Fibonacci numbers.
a(n) = F(n-1)^2 + F(n)^2 + F(n+1)^2 + F(n+2)^2 for n > 1, where F(n) is the n-th Fibonacci number (A000045). - Amiram Eldar, Jan 11 2022

A306823 a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n).

Original entry on oeis.org

1, 4, 27, 1048576, 45916502400000, 237376313799769806328950291431424, 18897697257047055734419223897702400000000000000000000000000000000000

Offset: 1

Emeric Deutsch, Apr 16 2019

nonn

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 second multiplicative Zagreb index of a simple connected graph is the product of deg(x)^(deg(x)) over all the vertices of the graph (see, for example, the I. Gutman reference, p. 16).
In the Maple program G = Sum_{n>=0} P[n]z^n is the generating function of the Lucas cubes according to size (coded by z) and vertex degrees (coded by t). See the Klavzar - Mollard - Petkovsek reference: l(x,y) on p. 1321 with different variables.

			a(2)=4 because the Lucas cube Lambda(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, a(2) = 1^1*1^1*2^2 = 4.
a(4)=1048576 because the Lucas cube Lambda(4) is a bouquet of tw 4-cycles, having 6 vertices of degree 2 and 1 vertex of degree 4; consequently, a(4) = (2^2)^6*4^4 = 2^12*v^4 = 1048576.

Alois P. Heinz, Table of n, a(n) for n = 1..11
I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institute ISSN 1840-4367, Vol. 1, 2011, 13-19.
S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A245960, A307580.

G:=(1+(1-t)*z+t^2*z^2+t*(1-t)*z^3-t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): Gser:=simplify(series(G, z=0,50)): for n from 0 to 45 do P[n]:=sort(coeff(Gser,z,n)) od: seq(product(j^(j*coeff(P[n],t,j)),j=1..n), n=1..7);

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A258121 Number of vertices of degree n in all Lucas cubes.

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