A305272 -id:A305272 - cp's OEIS frontend

A305269 a(n) = 120*2^n - 95.

25, 145, 385, 865, 1825, 3745, 7585, 15265, 30625, 61345, 122785, 245665, 491425, 982945, 1965985, 3932065, 7864225, 15728545, 31457185, 62914465, 125829025, 251658145, 503316385, 1006632865, 2013265825, 4026531745, 8053063585, 16106127265, 32212254625, 64424509345, 128849018785

Offset: 0

Emeric Deutsch, May 30 2018

a(n) is the number of vertices in the polyphenylene dendrimer G[n], defined pictorially in the Arif et al. reference (see Fig. 1, where G[2] is shown).

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.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A305270, A305271, A305272.

Maple
```
seq(120*2^n-95, n = 0..40);
```

Vec(5*(5 + 14*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 31 2018

From Colin Barker, May 31 2018: (Start)
G.f.: 5*(5 + 14*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
(End)

A305270 a(n) = 140*2^n - 112.

Original entry on oeis.org

28, 168, 448, 1008, 2128, 4368, 8848, 17808, 35728, 71568, 143248, 286608, 573328, 1146768, 2293648, 4587408, 9174928, 18349968, 36700048, 73400208, 146800528, 293601168, 587202448, 1174405008, 2348810128, 4697620368, 9395240848, 18790481808, 37580963728, 75161927568, 150323855248

Offset: 0

Emeric Deutsch, May 30 2018

a(n) is the number of edges in the polyphenylene dendrimer G[n], defined pictorially in the Arif et al. reference (see Fig. 1, where G[2] is shown).

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.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A305269, A305271, A305272.

Maple
```
seq(140*2^n-112, n = 0..40);
```

Vec(28*(1 + 3*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 31 2018

From Colin Barker, May 31 2018: (Start)
G.f.: 28*(1 + 3*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
(End)

A305271 a(n) = 680*2^n - 548.

Original entry on oeis.org

132, 812, 2172, 4892, 10332, 21212, 42972, 86492, 173532, 347612, 695772, 1392092, 2784732, 5570012, 11140572, 22281692, 44563932, 89128412, 178257372, 356515292, 713031132, 1426062812, 2852126172, 5704252892, 11408506332, 22817013212, 45634026972, 91268054492, 182536109532, 365072219612

Offset: 0

Emeric Deutsch, May 30 2018

a(n) is the first 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 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.
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, A305272.

Maple
```
seq(680*2^n-548, n = 0..40);
```

Vec(4*(33 + 104*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 31 2018

From Colin Barker, May 31 2018: (Start)
G.f.: 4*(33 + 104*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
(End)

cp's OEIS Frontend

A305269 a(n) = 120*2^n - 95.

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A305270 a(n) = 140*2^n - 112.

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Maple

PARI

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A305271 a(n) = 680*2^n - 548.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula