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A307208
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a(n) is the forgotten index of the Fibonacci cube Gamma(n).
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1
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2, 10, 52, 158, 466, 1192, 2914, 6722, 14972, 32286, 67914, 139824, 282754, 562970, 1105892, 2146846, 4124258, 7849496, 14815202, 27752338, 51632620, 95465502, 175508250, 320981472, 584214530, 1058602666, 1910305300, 3434059166, 6151218034, 10981579528
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OFFSET
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1,1
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COMMENTS
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The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's 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 Fibonacci cube Gamma(n) (see A245825).
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} T(n,k)*k^3 where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1).
G.f.: 2*x*(1 + x + 8*x^2 - 7*x^3 + 4*x^4 - 3*x^5 + 3*x^6) / (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>8.
(End)
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EXAMPLE
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a(2) = 10 because the Fibonacci cube Gamma(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.
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MAPLE
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T := (n, k) -> add(binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1), i=0..k):
seq(add(T(n, k)*k^3, k=1..n), n=1..30);
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PROG
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(PARI) T(n, k) = sum(i=0, k, binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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