A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).
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
Offset: 1
Keywords
Examples
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.
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.
Programs
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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);
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)
Comments