A304517 -id:A304517 - cp's OEIS frontend

A304518 a(n) = 68*2^n - 50 (n>=1).

86, 222, 494, 1038, 2126, 4302, 8654, 17358, 34766, 69582, 139214, 278478, 557006, 1114062, 2228174, 4456398, 8912846, 17825742, 35651534, 71303118, 142606286, 285212622, 570425294, 1140850638, 2281701326, 4563402702, 9126805454, 18253610958, 36507221966, 73014443982, 146028888014, 292057776078, 584115552206, 1168231104462

Offset: 1

Emeric Deutsch, May 15 2018

a(n) is the first Zagreb index of the nanostar dendrimer NS2[n] from the Madanshekaf et al. reference.
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 NS2[n] is M(NS2[n]; x,y) = 2*2^n*x*y^2 + (8*2^n - 5)*x^2*y^2 + (6*2^n - 6)*x^2*y^3.

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
A. Madanshekaf and M. Moradi, The first geometric-arithmetic index of some nanostar dendrimers, Iranian J. Math. Chemistry, 5, Supplement 1, 2014, s1-s6.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304517, A304519.

List([1..40],n->68*2^n-50); # Muniru A Asiru, May 15 2018

Maple
```
seq(68*2^n-50, n = 1 .. 40);
```

68*2^Range[50] - 50 (* Paolo Xausa, Jul 31 2024 *)
LinearRecurrence[{3,-2},{86,222},40] (* Harvey P. Dale, Jan 23 2025 *)

Vec(2*x*(43 - 18*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 15 2018

From Colin Barker, May 15 2018: (Start)
G.f.: 2*x*(43 - 18*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>2. (End)

A304519 a(n) = 72*2^n -56 (n>=1).

Original entry on oeis.org

88, 232, 520, 1096, 2248, 4552, 9160, 18376, 36808, 73672, 147400, 294856, 589768, 1179592, 2359240, 4718536, 9437128, 18874312, 37748680, 75497416, 150994888, 301989832, 603979720, 1207959496, 2415919048, 4831838152, 9663676360, 19327352776, 38654705608, 77309411272, 154618822600, 309237645256, 618475290568

Offset: 1

Emeric Deutsch, May 15 2018

a(n) is the second Zagreb index of the nanostar dendrimer NS2[n] from the Madanshekaf et al. reference.
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 NS2[n] is M(NS2[n]; x,y) = 2*2^n *x*y^2 + (8*2^n - 5)*x^2*y^2 + (6*2^n - 6)*x^2*y^3.

Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
A. Madanshekaf and M. Moradi, The first geometric-arithmetic index of some nanostar dendrimers, Iranian J. Math. Chemistry, 5, Supplement 1, 2014, s1-s6.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304517, A304518.

List([1..40],n->72*2^n-56); # Muniru A Asiru, May 15 2018

Maple
```
seq(72*2^n-56, n = 1 .. 40);
```

Vec(8*x*(11 - 4*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 15 2018

From Colin Barker, May 15 2018: (Start)
G.f.: 8*x*(11 - 4*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

cp's OEIS Frontend

A304518 a(n) = 68*2^n - 50 (n>=1).

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