A304164 -id:A304164 - cp's OEIS frontend

A304163 a(n) = 9n^2 - 3n + 1 with n>0.

7, 31, 73, 133, 211, 307, 421, 553, 703, 871, 1057, 1261, 1483, 1723, 1981, 2257, 2551, 2863, 3193, 3541, 3907, 4291, 4693, 5113, 5551, 6007, 6481, 6973, 7483, 8011, 8557, 9121, 9703, 10303, 10921, 11557, 12211, 12883, 13573, 14281

Offset: 1

Emeric Deutsch, May 09 2018

a(n) provides the number of vertices in the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).

			From _Andrew Howroyd_, May 09 2018: (Start)
Illustration of the order 1 graph:
    o---o
   / \ / \
  o---o---o
   \ / \ /
    o---o
The order 2 graph is composed of 7 such hexagons and in general the HcDN1(n) graph is constructed from a honeycomb graph with each hexagon subdivided into triangles.
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
S. Hayat, M. A. Malik, and M. Imran, Computing Topological Indices of Honeycomb Derived Networks, Romanian Journal for Information Science and Technology, Volume 18, Number 2, 2015, pages 144-165.
Leo Tavares, Illustration: Hexagonal Square Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304164.
First trisection of A002061 (without 1).
Cf. A003215, A000290.

[9*n^2-3*n+1 for n in 1:40] |> println # Bruno Berselli, May 10 2018

Maple
```
seq(9*n^2-3*n+1, n = 1 .. 40);
```

a(n) = 9*n^2-3*n+1; \\ Altug Alkan, May 09 2018

Vec(x*(7 + 10*x + x^2)/(1 - x)^3 + O(x^40)) \\ Colin Barker, May 23 2018

From Bruno Berselli, May 10 2018: (Start)
O.g.f.:  x*(7 + 10*x + x^2)/(1 - x)^3.
E.g.f.: -1 + (1 + 3*x)^2*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A003215(n-1) + 6*A000290(n). - Leo Tavares, Jul 21 2022

A304165 a(n) = 324n^2 - 336n + 102 (n >= 1).

Original entry on oeis.org

90, 726, 2010, 3942, 6522, 9750, 13626, 18150, 23322, 29142, 35610, 42726, 50490, 58902, 67962, 77670, 88026, 99030, 110682, 122982, 135930, 149526, 163770, 178662, 194202, 210390, 227226, 244710, 262842, 281622, 301050, 321126, 341850, 363222, 385242, 407910, 431226, 455190, 479802, 505062

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the first Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. manuscript).
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 HcDN1(n) is M(HcDN1(n);x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 -57n +30)x^6*y^6. - Emeric Deutsch, May 11 2018

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.
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304163, A304164, A304166.

List([1..40],n->324*n^2-336*n+102); # Muniru A Asiru, May 10 2018

Maple
```
seq(324*n^2-336*n+102,n=1..40);
```

Table[324n^2-336n+102,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{90,726,2010},40] (* Harvey P. Dale, Apr 12 2020 *)

a(n) = 324*n^2-336*n+102; \\ Altug Alkan, May 09 2018

Vec(6*x*(15 + 76*x + 17*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: 6*x*(15 + 76*x + 17*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 6*(exp(x)*(17 - 2*x + 54*x^2) - 17). - Stefano Spezia, Apr 15 2023

A304166 a(n) = 972n^2 - 1224n + 414 with n > 0.

Original entry on oeis.org

162, 1854, 5490, 11070, 18594, 28062, 39474, 52830, 68130, 85374, 104562, 125694, 148770, 173790, 200754, 229662, 260514, 293310, 328050, 364734, 403362, 443934, 486450, 530910, 577314, 625662, 675954, 728190, 782370, 838494, 896562, 956574, 1018530, 1082430, 1148274, 1216062, 1285794, 1357470, 1431090, 1506654

Offset: 1

Emeric Deutsch, May 09 2018

a(n) provides the second Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).
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 HcDN1(n) is M(HcDN1(n); x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 - 57n + 30)x^6*y^6. - Emeric Deutsch, May 11 2018

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.
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304163, A304164, A304165.

Maple
```
seq(972*n^2-1224*n+414, n = 1 .. 40);
```

a(n) = 972*n^2-1224*n+414; \\ Altug Alkan, May 09 2018

Vec(18*x*(9 + 76*x + 23*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: 18*x*(9 + 76*x + 23*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 18*(exp(x)*(23 - 14*x + 54*x^2) - 23). - Stefano Spezia, Apr 15 2023

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A304163 a(n) = 9*n^2 - 3*n + 1 with n>0.

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A304165 a(n) = 324*n^2 - 336*n + 102 (n >= 1).

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A304166 a(n) = 972*n^2 - 1224*n + 414 with n > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

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Maple

PARI

PARI

Formula

A304163 a(n) = 9n^2 - 3n + 1 with n>0.

A304165 a(n) = 324n^2 - 336n + 102 (n >= 1).

A304166 a(n) = 972n^2 - 1224n + 414 with n > 0.