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%I #17 Jan 12 2024 10:03:47
%S 3,18,2925,11216538648,627285206516110230354416268831,
%T 109715796815760578436090875708748277077073796614051376195149103817368827024587948919162326
%N Number of maximum independent vertex sets in the n-Hanoi graph.
%H Pontus von Brömssen, <a href="/A297536/b297536.txt">Table of n, a(n) for n = 1..8</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumIndependentVertexSet.html">Maximum Independent Vertex Set</a>
%t (* Except for one of the initial values, this program is identical to the program for A288490. *)
%t {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}] @@ # &, {0, 1, 0, 0}, 4] (* _Pontus von Brömssen_, Mar 14 2020 *)
%o (PARI)
%o \\ Except for one of the initial values, this program is identical to the program by _Andrew Howroyd_ for A288490.
%o 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]}
%o a(n) = {my(v); v=[0, 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])} \\ _Pontus von Brömssen_, Mar 14 2020
%o (Python)
%o from itertools import islice
%o def A297536_gen(): # generator of terms
%o f,g,h,p = 0,1,0,0
%o while True:
%o yield f+3*(g+h)+p
%o a, b = f+(g<<1), g+(h<<1)
%o 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)
%o A297536_list = list(islice(A297536_gen(),6)) # _Chai Wah Wu_, Jan 11 2024
%Y Cf. A288490 (independent vertex sets in the n-Hanoi graph).
%Y Cf. A321249 (maximal independent vertex sets in the n-Hanoi graph).
%Y Cf. A288839 (chromatic polynomials of the n-Hanoi graph).
%Y Cf. A193233 (chromatic polynomial with highest coefficients first).
%Y Cf. A137889 (directed Hamiltonian paths in the n-Hanoi graph).
%Y Cf. A286017 (matchings in the n-Hanoi graph).
%Y Cf. A193136 (spanning trees of the n-Hanoi graph).
%Y Cf. A288796 (undirected paths in the n-Hanoi graph).
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Dec 31 2017
%E More terms from _Pontus von Brömssen_, Mar 14 2020
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