A153796 -id:A153796 - cp's OEIS frontend

A153795 5 times octagonal numbers: a(n) = 5n(3*n-2).

0, 5, 40, 105, 200, 325, 480, 665, 880, 1125, 1400, 1705, 2040, 2405, 2800, 3225, 3680, 4165, 4680, 5225, 5800, 6405, 7040, 7705, 8400, 9125, 9880, 10665, 11480, 12325, 13200, 14105, 15040, 16005, 17000, 18025, 19080, 20165, 21280

Offset: 0

Omar E. Pol, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A153794, A153796.

Table[5 * n * (3 * n - 2) , {n, 0, 25}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 40}, 25] (* G. C. Greubel, Aug 28 2016 *)

a(n)=5*n*(3*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 15*n^2 - 10*n = A000567(n)*5.
a(n) = 30*n + a(n-1) - 25 for n > 0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 5*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 5*x*(1 + 3*x)*exp(x). (End)

A153797 7 times octagonal numbers: a(n) = 7n(3*n-2).

Original entry on oeis.org

0, 7, 56, 147, 280, 455, 672, 931, 1232, 1575, 1960, 2387, 2856, 3367, 3920, 4515, 5152, 5831, 6552, 7315, 8120, 8967, 9856, 10787, 11760, 12775, 13832, 14931, 16072, 17255, 18480, 19747, 21056, 22407, 23800, 25235, 26712, 28231

Offset: 0

Omar E. Pol, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567 (octagonal numbers), A153796, A153808.

Magma
```
[ 7*n*(3*n-2): n in [0..40] ];
```

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,7,8!,42}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Table[7*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,7,56},25] (* G. C. Greubel, Aug 29 2016 *)
7*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, May 16 2025 *)

a(n)=7*n*(3*n-2) \\ Charles R Greathouse IV, Aug 29 2016

a(n) = 21*n^2 - 14*n = 7*A000567(n).
a(n) = a(n-1) + 42*n - 35 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 7*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 7*x*(1 + 3*x)*exp(x). (End)

A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

Original entry on oeis.org

6, 6, 3, 30, 48, 57, 54, 45, 30, 12, 72, 126, 165, 186, 195, 186, 168, 138, 102, 66, 27, 132, 240, 327, 390, 435, 456, 462, 444, 414, 366, 309, 246, 177, 114, 48, 210, 390, 543, 666, 765, 834, 882, 900, 900, 870, 825, 756, 675, 582, 480, 378, 270, 174, 75

Offset: 1

Emeric Deutsch, Aug 31 2012

The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
Row n contains 4n-1 entries.
T(n,1) = 9n^2-3n = A152743(n).
T(n,2) = 6n(3n-2)= A153796(n).
T(n,3) = 3(9n^2-9n+1)= 3*A069131(n) (for n>5 this is a conjecture).
T(n,2n) = n(7n^2-1) = 6*A004126(n) (for n>5 this is a conjecture).
T(n,4n-2) = 6(n^2+n-1) = 6*A028387(n-1) (for n>5 this is a conjecture).
T(n,4n-1) = 3n^2 = A033428(n) (for n>5 this is a conjecture).
Sum(k*T(n,k), k>=1) = A143366(n).

S. Klavzar, A bird's eye view of the cut method and a survey of its applications in chemical graph theory, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.
Bo-Yin Yang and Yeong-Nan Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91, 2004.
P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.

Cf. A143366, A214580, A152743, A153796, A069131, A004126, A028387, A033428.

The entries have been obtained by using the Maple Graph Theory package for finding the distance matrix of each of the five graphs H(n) (n=1,2,3,4,5). The given Maple program yields the Wiener polynomial of H(2) (having as coefficients the entries in row 2).

cp's OEIS Frontend

A153795 5 times octagonal numbers: a(n) = 5n(3*n-2).

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A153797 7 times octagonal numbers: a(n) = 7n(3*n-2).

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Magma

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A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

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cp's OEIS Frontend

A153795 5 times octagonal numbers: a(n) = 5*n*(3*n-2).

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Author

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Crossrefs

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Mathematica

PARI

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A153797 7 times octagonal numbers: a(n) = 7*n*(3*n-2).

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Author

Keywords

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

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Formula

A153795 5 times octagonal numbers: a(n) = 5n(3*n-2).

A153797 7 times octagonal numbers: a(n) = 7n(3*n-2).