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A307307 a(n) is the forgotten index of the Lucas cube Lambda(n). 2
0, 10, 30, 112, 300, 840, 2044, 4864, 10944, 23960, 50908, 105840, 215748, 432656, 855240, 1669568, 3223404, 6162552, 11678540, 21957440, 40988976, 76019944, 140155100, 256995936, 468887700, 851538064, 1539858168, 2773522192, 4977094956, 8900629800 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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 forgotten topological index of a simple connected graph is the sum of the cubes of its vertex degrees.
In the Maple program T(n,k) gives the number of vertices of degree k in the Lucas cube Lambda(n).
LINKS
B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184-1190, 2015.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
FORMULA
Conjectures from Colin Barker, Apr 02 2019: (Start)
G.f.: 2*x^2*(5 - 5*x + 6*x^2 - 4*x^3 + 27*x^4 - 25*x^5 - 6*x^6 + 9*x^8 + 3*x^9) / (1 - x - x^2)^4.
a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n>11.
(End)
EXAMPLE
a(2) = 10 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, the forgotten index is 1^3 + 1^3 + 2^3 = 10.
MAPLE
G:=(1+(1-t)*z + t^2*z^2 + (1-t)*t*z^3 - t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): M:=expand(series(G, z=0, 40)): T:=(n, k)->coeff(coeff(M, z, n), t, k): FI:=n->add(T(n, k)*k^3, k=0..n): seq(FI(n), n=1..30);
CROSSREFS
Sequence in context: A057344 A115134 A245428 * A185829 A301664 A335803
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 02 2019
STATUS
approved

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Last modified July 12 10:18 EDT 2024. Contains 374244 sequences. (Running on oeis4.)