A338293 -id:A338293 - cp's OEIS frontend

A228607 a(n) is the number of independent vertex subsets (i.e., the Merrifield-Simmons index) of the triangulane T[n] defined in the Khalifeh et al. and Deutsch et al. references.

54, 9450, 286679250, 263199759084281250, 221721055245240563933498289550781250, 157320497971930517299046640166039915248640240419548633694648742675781250

Offset: 1

Emeric Deutsch, Dec 17 2013

nonn

R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.

Emeric Deutsch, Table of n, a(n) for n = 1..9
E. Deutsch, S. Klavzar, Computing the Hosoya polynomial of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem., 70, 2013, 627-644.
M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, Computing Wiener and Kirchhoff indices of a triangulane, Indian J. Chemistry, 47A, 2008, 1503-1507.
H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20, 1982, 16-21.
Wikipedia, Triangulane

c[1] := 1: d[1] := 3: for n from 2 to 10 do c[n] := d[n-1]^2; d[n] := 2*c[n-1]*d[n-1]+d[n-1]^2 end do: seq(d[n]^3+3*c[n]*d[n]^2, n = 1 .. 7);

a(n) = d(n)^3 + 3*c(n)*d(n)^2, where c(1) = 1, d(1) = 3, c(n) = d(n-1)^2, d(n) = 2*c(n-1)*d(n-1) + d(n-1)^2 for n>=2.
If we replace the initial conditions for c and d by c[1] = x and d[1] = 1 + 2x, respectively, and the first equation by c[n] = x*d[n-1]^2, then a(n) will yield the independence polynomial of the triangulane T(n). For example, for n=2 one finds 1 + 21x + 180x^2 + 816x^3 + 2112x^4 + 3120x^5 + 2432x^6 + 768x^7 (checked with the Maple Graph Theory package).
d(n) = A338293(n+1). - R. J. Mathar, Jul 22 2022

A338294 Decimal expansion of the growth power of the number of matchings in the complete binary tree.

Original entry on oeis.org

1, 5, 3, 7, 1, 7, 6, 7, 1, 7, 1, 8, 2, 3, 5, 7, 9, 4, 9, 5, 9, 0, 1, 4, 0, 3, 2, 8, 9, 5, 5, 2, 2, 1, 6, 0, 2, 5, 0, 1, 5, 0, 8, 0, 9, 3, 4, 3, 2, 3, 6, 1, 7, 3, 2, 6, 5, 1, 0, 2, 0, 0, 2, 9, 5, 0, 1, 4, 0, 0, 8, 8, 2, 7, 0, 8, 8, 8, 1, 1, 2, 4, 3, 5, 3, 2, 9

Offset: 1

Kevin Ryde, Oct 21 2020

The number of matchings in the complete binary tree of n rows is A338293(n). It grows as A338293(n) ~ (1/2)*C^(2^n) where C is the present constant. See A338293 on how log(C) = A242049 follows from the number of matchings as a product of Jacobsthal numbers.

			1.537176717...

Kevin Ryde, vpar examples/complete-binary-matchings.gp calculations and code in PARI/GP.

Cf. A242049, A338293.

Equals exp(A242049).

Equals lim_{n->oo} A338293(n)^(1/2^n).

A377936 Number of matchings in the complete planted binary tree with 2^n leaves.

Original entry on oeis.org

2, 4, 24, 720, 712800, 666860040000, 597568733024952150000000, 474258018883889933710067708314342382812500000000

Offset: 0

Atabey Kaygun, Nov 11 2024

nonn

A planted binary tree has an initial root node with 1 child. The root is not considered to be a leaf. All internal nodes have degree 3. The total number of nodes is 2*n.

			The initial graphs for n=0..2 are:
      o        o                 o
      |        |                 |
      o        o                 o
             /   \             /   \
            o     o           o     o
                             / \   / \
                            o   o o   o

Andrew Howroyd, Table of n, a(n) for n = 0..11
Atabey Kaygun Hosoya Index of Balanced Binary Trees.
Eric Weisstein's World of Mathematics, Independent Edge Set.
Eric Weisstein's World of Mathematics, Matching.

Cf. A338293.

lista(n)={my(u=vector(n), v=vector(n)); u[1]=v[1]=1; for(n=1, #u-1, u[n+1]=v[n]^2; v[n+1]=u[n+1] + 2*v[n]*u[n]); v+u} \\ Andrew Howroyd, Nov 14 2024

a(n) = u(n) + v(n) where u(n) = v(n-1)^2 and v(n) = v(n-1)^2 + 2*v(n-1)*u(n-1) with u(1) = v(1) = 1. - Andrew Howroyd, Nov 14 2024

a(5) onwards from Andrew Howroyd, Nov 14 2024

cp's OEIS Frontend

A228607 a(n) is the number of independent vertex subsets (i.e., the Merrifield-Simmons index) of the triangulane T[n] defined in the Khalifeh et al. and Deutsch et al. references.

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A338294 Decimal expansion of the growth power of the number of matchings in the complete binary tree.

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A377936 Number of matchings in the complete planted binary tree with 2^n leaves.

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