A245825 -id:A245825 - cp's OEIS frontend

A306967 a(n) is the first Zagreb index of the Fibonacci cube Gamma(n).

2, 6, 22, 54, 132, 292, 626, 1290, 2594, 5102, 9864, 18792, 35362, 65838, 121454, 222246, 403788, 728972, 1308562, 2336946, 4154170, 7353310, 12965904, 22781520, 39897410, 69662166, 121292998, 210642966, 364928532, 630794356

Offset: 1

Emeric Deutsch, Mar 26 2019

nonn

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 first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
In the Maple program, T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).

			a(2) = 6 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 Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).

S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A245825.

T:=(n,k) -> sum(binomial(n - 2*i, k - i) * binomial(i + 1, n - k - i + 1), i = 0..k): seq(add(T(n, k)*k^2, k=1..n), n=1..30);

a(n) = Sum_{k=1..n} T(n,k)*k^2, where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i) * binomial(i+1, n-k-i+1).
Conjectures from Colin Barker, Mar 28 2019: (Start)
G.f.: 2*x*(1 + 2*x^2 - x^3) / (1 - x - x^2)^3.
a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>6.
(End)

A307157 a(n) is the Narumi-Katayama index of the Fibonacci cube Gamma(n).

Original entry on oeis.org

1, 2, 24, 1152, 1399680, 290237644800, 520105859481600000000, 3435834286784202670080000000000000000, 3045775242579858715944293498880000000000000000000000000000000000

Offset: 1

Emeric Deutsch, Mar 27 2019

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 Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.

			a(2)=2 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 Narumi-Katayama index is 1*1*2=2.

I. Gutman and M. Ghorbani, Some properties of the Narumi-Katayama index, Applied Mathematics Letters, Vol. 25, No. 10 (2012), 1435-1438.
S. Klavžar, Structure of Fibonacci cubes: a survey, Journal of Combinatorial Optimization, Vol. 25, No. 4 (2013), 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A000045, A245825.

T := (n,k) -> add(binomial(n-2*i,k-i)*binomial(i+1,n-k-i+1), i=0..k):
seq(mul(j^T(n,j), j=1..n), n=1..10);

a(n) = Product_{k=1..n} k^T(n, k), where T(n, k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1). T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).

A307580 a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).

Original entry on oeis.org

1, 4, 1728, 191102976, 137105941502361600000, 27038645743755029502156994133360640000000000, 645557379413314860145212937623335060473992141864960000000000000000000000000000000000000000

Offset: 1

Emeric Deutsch, Apr 15 2019

nonn

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 second multiplicative Zagreb index of a simple connected graph is the product of deg(x)^(deg(x)) over all the vertices x of the graph (see, for example, the I. Gutman reference, p. 16).
In the Maple program, T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825 and the KLavzar - Mollard - Petkovsek reference).

			a(2) = 4 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, a(2) = 1^1*1^1*2^2 = 4.
a(4) = 191102976 because the Fibonacci cube Gamma(4) has 5 vertices of degree 2, 2 vertices of degree 3, and 1 vertex of degree 4; consequently, a(4) = (2^2)^5*(3^3)^2*4^4 = 191102976.

Alois P. Heinz, Table of n, a(n) for n = 1..10
I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institute ISSN 1840-4367, Vol. 1, 2011, 13-19.
S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A245825.

T := (n,k)-> add(binomial(n-2*i,k-i)*binomial(i+1,n-k-i+1), i=0..k):
seq(mul(k^(k*T(n,k)), k=1..n), n=1..7);

a(n) = Product_{k=1..n} k^(k*T(n,k)), where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1).

A307208 a(n) is the forgotten index of the Fibonacci cube Gamma(n).

Original entry on oeis.org

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

Offset: 1

Emeric Deutsch, Mar 28 2019

nonn

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).

			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.

B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184-1190, 2015.
S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A245825, A306967, A307157.

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);

T(n,k) = sum(i=0, k, binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1));
a(n) = sum(k=1, n, T(n,k)*k^3); \\ Michel Marcus, Mar 30 2019

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).
Conjectures from Colin Barker, Mar 29 2019: (Start)
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)

cp's OEIS Frontend

A306967 a(n) is the first Zagreb index of the Fibonacci cube Gamma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A307157 a(n) is the Narumi-Katayama index of the Fibonacci cube Gamma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A307580 a(n) is the second multiplicative Zagreb index of the Fibonacci cube Gamma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A307208 a(n) is the forgotten index of the Fibonacci cube Gamma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula