Emeric Deutsch - cp's OEIS frontend

A306823 a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n).

1, 4, 27, 1048576, 45916502400000, 237376313799769806328950291431424, 18897697257047055734419223897702400000000000000000000000000000000000

Offset: 1

Emeric Deutsch, Apr 16 2019

nonn

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, 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 of the graph (see, for example, the I. Gutman reference, p. 16).
In the Maple program G = Sum_{n>=0} P[n]z^n is the generating function of the Lucas cubes according to size (coded by z) and vertex degrees (coded by t). See the Klavzar - Mollard - Petkovsek reference: l(x,y) on p. 1321 with different variables.

			a(2)=4 because the Lucas cube Lambda(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)=1048576 because the Lucas cube Lambda(4) is a bouquet of tw 4-cycles, having 6 vertices of degree 2 and 1 vertex of degree 4; consequently, a(4) = (2^2)^6*4^4 = 2^12*v^4 = 1048576.

Alois P. Heinz, Table of n, a(n) for n = 1..11
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. A245960, A307580.

G:=(1+(1-t)*z+t^2*z^2+t*(1-t)*z^3-t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): Gser:=simplify(series(G, z=0,50)): for n from 0 to 45 do P[n]:=sort(coeff(Gser,z,n)) od: seq(product(j^(j*coeff(P[n],t,j)),j=1..n), n=1..7);

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).

A307559 a(n) = floor(n/3)(n - floor(n/3))(n - floor(n/3) - 1).

Original entry on oeis.org

0, 0, 2, 6, 12, 24, 40, 60, 90, 126, 168, 224, 288, 360, 450, 550, 660, 792, 936, 1092, 1274, 1470, 1680, 1920, 2176, 2448, 2754, 3078, 3420, 3800, 4200, 4620, 5082, 5566, 6072, 6624, 7200, 7800, 8450, 9126, 9828, 10584, 11368, 12180, 13050, 13950, 14880, 15872

Offset: 1

Emeric Deutsch, Apr 14 2019

a(n) is an upper bound for the irregularity of a graph with n vertices (see Theorem 3.2 of the Tavakoli et al. reference).

			a(4) = floor(4/3)*(4 - floor(4/3))*(4-floor(4/3)-1) = 1*3*2 = 6.

M. Tavakoli, F. Rahbarnia, M. Mirzavaziri, A. R. Ashrafi, and I. Gutman, Extremely irregular graphs, Kragujevac J. Math., 37 (1), 2013, 135-139.
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A200067.

a:=n->floor(n/3)*(n-floor(n/3))*(n-floor(n/3)-1): seq(a(n), n=1..50);

a(n) = 2*A200067(n).

G.f.: 2*x^3*(1+x)*(1+x^2) / ( (1+x+x^2)^2*(x-1)^4 ). - R. J. Mathar, Jul 22 2022

A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).

Original entry on oeis.org

0, 10, 30, 112, 300, 840, 2044, 4864, 10944, 23960, 50908, 105840, 215748, 432656, 855240, 1669568, 3223404, 6162552, 11678540, 21957440, 40988976, 76019944, 140155100, 256995936, 468887700, 851538064, 1539858168, 2773522192, 4977094956, 8900629800

Offset: 1

Emeric Deutsch, Apr 02 2019

nonn

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, 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 Lucas cube Lambda(n).

			a(2) = 10 because the Lucas cube Lambda(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, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A307181, A307208, A307212.

G:=(1+(1-t)*z + t^2*z^2 + (1-t)*t*z^3 - t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): M:=expand(series(G,z=0,40)): T:=(n,k)->coeff(coeff(M,z,n),t,k): FI:=n->add(T(n,k)*k^3, k=0..n): seq(FI(n), n=1..30);

Conjectures from Colin Barker, Apr 02 2019: (Start)
G.f.: 2*x^2*(5 - 5*x + 6*x^2 - 4*x^3 + 27*x^4 - 25*x^5 - 6*x^6 + 9*x^8 + 3*x^9) / (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>11.
(End)

A307212 a(n) is the Narumi-Katayama index of the Lucas cube Lambda(n).

Original entry on oeis.org

0, 2, 3, 256, 38880, 1289945088, 42855402240000000, 605828739547255327948800000000, 13263549731442762279026688000000000000000000000000000, 1334793240853871268746431553848403294648071618560000000000000000000000000000000000000000000

Offset: 1

Emeric Deutsch, Mar 28 2019

nonn

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, 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 Lucas cube Lambda(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, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A307157, A307181, A307307.

G := (1+(1-y)*x+x^2*y^2+(1-y)*x^3*y-(1-y)^2*x^4*y)/((1-x*y)*(1-x^2*y)-x^3*y):
g := expand(series(G, x=0, 40)): T := (n, k) -> coeff(coeff(g, x, n), y, k):
a := n -> mul(k^T(n, k), k=0..n): lprint(seq(a(n), n=1..10));

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)

A307181 a(n) is the first Zagreb index of the Lucas cube Lambda(n).

Original entry on oeis.org

0, 6, 12, 40, 90, 216, 462, 976, 1980, 3940, 7678, 14736, 27898, 52220, 96780, 177824, 324258, 587304, 1057350, 1893320, 3373692, 5984924, 10574342, 18613920, 32654450, 57106036, 99576972, 173166616, 300385770, 519849720

Offset: 1

Emeric Deutsch, Mar 28 2019

nonn

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, 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 Lucas cube Lambda(n).

			a(2) = 6 because the Lucas cube Lambda(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently the first Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).

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. A306967, A307307, A307212

L:=(1+(1-y)*x + x^2*y^2 + (1-y)*x^3*y-(1-y)^2*x^4*y)/((1-x*y)*(1-x^2*y)-x^3*y): M:=expand(series(L,x=0,40)): T:=(n,k)->coeff(coeff(M,x,n),y,k): Z1:=n->add(T(n,k)*k^2, k=0..n): seq(Z1(n), n=1..35);

Conjectures from Colin Barker, Mar 28 2019: (Start)
G.f.: 2*x^2*(3 - 3*x + 2*x^2 + 3*x^4 - 2*x^5 - x^6) / (1 - x - x^2)^3.
a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>8.
(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).

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

Original entry on oeis.org

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)

A305272 a(n) = 836*2^n - 676.

Original entry on oeis.org

160, 996, 2668, 6012, 12700, 26076, 52828, 106332, 213340, 427356, 855388, 1711452, 3423580, 6847836, 13696348, 27393372, 54787420, 109575516, 219151708, 438304092, 876608860, 1753218396, 3506437468, 7012875612, 14025751900, 28051504476, 56103009628, 112206019932, 224412040540, 448824081756

Offset: 0

Emeric Deutsch, May 30 2018

a(n) is the second Zagreb index of the polyphenylene dendrimer G[n], defined pictorially in the Arif et al. reference (see Fig. 1, where G[2] is shown).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of the polyphenylene dendrimer G[n] is M(G[n]; x, y) = (56*2^n - 40)*x^2*y^2 + (48*2^n - 40)*x^2*y^3 +(36* 2^n - 36)*x^3*y^3 + 4*x^3 *y^4.

Colin Barker, Table of n, a(n) for n = 0..1000
N. E. Arif, Roslan Hasni and Saeid Alikhani, Fourth order and fourth sum connectivity indices of polyphenylene dendrimers, J. Applied Science, 12 (21), 2012, 2279-2282.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A305269, A305270, A305271.

Maple
```
seq(836*2^n-676, n = 0..40);
```

836*2^Range[0,40]-676 (* or  *) LinearRecurrence[{3,-2},{160,996},40] (* Harvey P. Dale, Jun 19 2021 *)

Vec(4*(40 + 129*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 31 2018

From Colin Barker, May 31 2018: (Start)
G.f.: 4*(40 + 129*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
(End)

cp's OEIS Frontend

User: Emeric Deutsch

A306823 a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n).

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

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A307559 a(n) = floor(n/3)*(n - floor(n/3))*(n - floor(n/3) - 1).

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A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).

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A307212 a(n) is the Narumi-Katayama index of the Lucas cube Lambda(n).

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

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A307181 a(n) is the first Zagreb index of the Lucas cube Lambda(n).

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

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A307559 a(n) = floor(n/3)(n - floor(n/3))(n - floor(n/3) - 1).