A304170 -id:A304170 - cp's OEIS frontend

A304169 a(n) = 16*3^n + 2^(n+1) - 26 (n>=1).

26, 126, 422, 1302, 3926, 11766, 35222, 105462, 315926, 946806, 2838422, 8511222, 25525526, 76560246, 229648022, 688878582, 2066504726, 6199252086, 18597232022, 55790647542, 167369845526, 502105342326, 1506307638422, 4518906138102, 13556684859926, 40669987470966

Offset: 1

Emeric Deutsch, May 11 2018

For n>=2, a(n) is the first Zagreb index of the Sierpinski Gasket Rhombus graph SR[n] (see the Antony Xavier 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 the Sierpinski Gasket Rhombus graph SR[n] is M(SR[n]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 +2*x^3*y^6 + (2*3^n - 3*2^n - 4)*x^4*y^4 + (2^{n+1} - 4)*x^4*y^6 + (2^{n-1} - 2)*x^6*y^6.

Colin Barker, Table of n, a(n) for n = 1..1000
D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, Topological properties of Sierpinski Gasket Rhombus graphs, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
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 (6,-11,6).

Cf. A304170.

Maple
```
seq(16*3^n+2^(n+1)-26, n = 1 .. 30);
```

Vec(2*x*(13 - 15*x - 24*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 11 2018

From Colin Barker, May 11 2018: (Start)
G.f.: 2*x*(13 - 15*x - 24*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
(End)

A304167 a(n) = 3^n - 2^(n-1) + 2 (n>=1).

Original entry on oeis.org

4, 9, 25, 75, 229, 699, 2125, 6435, 19429, 58539, 176125, 529395, 1590229, 4774779, 14332525, 43013955, 129074629, 387289419, 1161999325, 3486260115, 10459304629, 31378962459, 94138984525, 282421147875, 847271832229, 2541832273899, 7625530376125, 22876658237235, 68630108929429, 205890595223739

Offset: 1

Emeric Deutsch, May 10 2018

For n>=2, a(n) is the number of vertices of the Sierpinski Gasket Rhombus graph SR(n) (see Theorem 2.1 in the D. Antony Xavier et al. reference).

Colin Barker, Table of n, a(n) for n = 1..1000
D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, Topological properties of Sierpinski Gasket Rhombus graphs, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A304168, A304169, A304170.

List([1..40],n->3^n-2^(n-1)+2); # Muniru A Asiru, May 10 2018

Maple
```
seq(3^n-2^(n-1)+2, n = 1 .. 40);
```

Vec(x*(4 - 15*x + 15*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: x*(4 - 15*x + 15*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
(End)
a(n) = A083313(n)+2. - R. J. Mathar, Jul 24 2022

A304168 a(n) = 2*3^n - 2^(n-1) (n>=1).

Original entry on oeis.org

5, 16, 50, 154, 470, 1426, 4310, 12994, 39110, 117586, 353270, 1060834, 3184550, 9557746, 28681430, 86060674, 258214790, 774709906, 2324260790, 6973044514, 20919657830, 62760022066, 188282163350, 564850684354, 1694560441670, 5083698102226, 15251127861110, 45753450692194, 137260486294310, 411781727318386

Offset: 1

Emeric Deutsch, May 10 2018

For n>=2, a(n) is the number of edges of the Sierpinski Gasket Rhombus graph SR(n) (see Theorem 2.1 in the D. Antony Xavier et al. reference).

Colin Barker, Table of n, a(n) for n = 1..1000
D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, Topological properties of Sierpinski Gasket Rhombus graphs, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A304167, A304169, A304170.

List([1..35],n->2*3^n-2^(n-1)); # Muniru A Asiru, May 10 2018

Maple
```
seq(2*3^n-2^(n-1), n = 1 .. 40);
```

Vec(x*(5 - 9*x) / ((1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: x*(5 - 9*x) / ((1 - 2*x)*(1 - 3*x)).
a(n) = 5*a(n-1) - 6*a(n-2) for n>2.
(End)

cp's OEIS Frontend

A304169 a(n) = 16*3^n + 2^(n+1) - 26 (n>=1).

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A304167 a(n) = 3^n - 2^(n-1) + 2 (n>=1).

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Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

PARI

Formula

A304168 a(n) = 2*3^n - 2^(n-1) (n>=1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

PARI

Formula