A288490 Number of independent vertex sets and vertex covers in the n-Hanoi graph.

4, 52, 108144, 967067994163264, 691513106932053164262669026747190128930258944

Offset: 1

Author

Eric W. Weisstein, Jun 16 2017

Keywords

• nonn

Comments

Term a(6) has 135 decimal digits and a(7) has 404 decimal digits. - Andrew Howroyd, Jun 19 2017

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..7
• Hanlin Chen, Renfang Wu, Guihua Huang, and Hanyuan Deng, Independent Sets on the Towers of Hanoi Graphs, Ars Mathematica Contemporanea, volume 12, number 2, 2017, pages 247-260. Section 3, i_n = a(n).
• Eric Weisstein's World of Mathematics, Hanoi Graph
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Eric Weisstein's World of Mathematics, Vertex Cover

Crossrefs

Cf. A297536 (maximum independent vertex sets in the n-Hanoi graph).

Cf. A321249 (maximal independent vertex sets in the n-Hanoi graph).

Cf. A288839 (chromatic polynomials of the n-Hanoi graph).

Cf. A193233 (chromatic polynomial with highest coefficients first).

Cf. A137889 (directed Hamiltonian paths in the n-Hanoi graph).

Cf. A286017 (matchings in the n-Hanoi graph).

Cf. A193136 (spanning trees of the n-Hanoi graph).

Cf. A288796 (undirected paths in the n-Hanoi graph).

Programs

Mathematica

{1, 3, 3, 1} . # & /@ NestList[Function[{h, i, j, k}, {h^3 + 6 h^2 i + 9 h i^2 + 3 h^2 j + 2 i^3 + 6 h i j, h^2 i + 4 h i^2 + 2 h^2 j + h^2 k + 8 h i j + 3 i^3 + 4 i^2 j + 2 h j^2 + 2 h i k, h i^2 + 4 h i j + 2 i^3 + 7 i^2 j + 2 h i k + 3 h j^2 + 4 i j^2 + 2 i^2 k + 2 h j k, i^3 + 6 i^2 j + 9 i j^2 + 3 i^2 k + 2 j^3 + 6 i j k}] @@ # &, {1, 1, 0, 0}, 4]

PARI

\\ Here h0..h3 is independent sets with 0..3 of the 3 apex vertices occupied.
Next(h0,h1,h2,h3) = {[h0^3 + 6*h0^2*h1 + 9*h0*h1^2 + 3*h0^2*h2 + 2*h1^3 + 6*h0*h1*h2, h0^2*h1 + 4*h0*h1^2 + 2*h0^2*h2 + h0^2*h3 + 8*h0*h1*h2 + 3*h1^3 + 4*h1^2*h2 + 2*h0*h2^2 + 2*h0*h1*h3, h0*h1^2 + 4*h0*h1*h2 + 2*h1^3 + 7*h1^2*h2 + 2*h0*h1*h3 + 3*h0*h2^2 + 4*h1*h2^2 + 2*h1^2*h3 + 2*h0*h2*h3, h1^3 + 6*h1^2*h2 + 9*h1*h2^2 + 3*h1^2*h3 + 2*h2^3 + 6*h1*h2*h3]}
a(n) = {my(v);v=[1,1,0,0]; for(i=2,n,v=Next(v[1],v[2],v[3],v[4])); v[1]+v[4]+3*(v[2]+v[3])} \\ Andrew Howroyd, Jun 20 2017

Python

from itertools import islice
def A288490_gen(): # generator of terms
    f,g,h,p = 1,1,0,0
    while True:
        yield f+3*(g+h)+p
        a, b = f+(g<<1), g+(h<<1)
        f,g,h,p = a*(f*(a+(b<<1)-h)+g**2), f*(p*a+b*(a+(g<<1))+2*h**2)+g**2*(g+(b<<1)), f*(g*(b+(h<<1))+3*h**2)+g*(g*((b<<1)+3*h)+(h<<1)**2)+p*(f*b+g*a), b*(g*(3*p+b+(h<<1))+h**2)
A288490_list = list(islice(A288490_gen(),5)) # Chai Wah Wu, Jan 11 2024

Extensions

a(5) from Andrew Howroyd, Jun 19 2017