A014642 -id:A014642 - cp's OEIS frontend

A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).

27, 144, 351, 648, 1035, 1512, 2079, 2736, 3483, 4320, 5247, 6264, 7371, 8568, 9855, 11232, 12699, 14256, 15903, 17640, 19467, 21384, 23391, 25488, 27675, 29952, 32319, 34776, 37323, 39960, 42687, 45504, 48411, 51408, 54495, 57672, 60939, 64296, 67743, 71280, 74907

Offset: 1

Emeric Deutsch, Sep 30 2010

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

			a(1)=27 because in D(6,1)=C_6 we have 6 distances equal to 1, 6 distances equal to 2, and 3 di stances equal to 3.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014642, A033991, A147874, A180579, A180867.

Maple
```
seq(9*n*(5*n-2), n = 1 .. 40);
```

a(n)=9*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A180867(6,n).
a(n) = 9*n*(5*n-2).
The Wiener polynomial of the graph D(6,n) is (1/2)nt(t^2+2t+2)((n-1)t^3+2(n-1)t^2+2(n-1)t+6).
G.f.: -9*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 31 2012
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 9*exp(x)*x*(3 + 5*x).
a(n) = 9*A147874(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

Duplicated a(38) removed by Sean A. Irvine, Apr 14 2025

A180579 The Wiener index of the Dutch windmill graph D(5,n) (n>=1).

Original entry on oeis.org

15, 78, 189, 348, 555, 810, 1113, 1464, 1863, 2310, 2805, 3348, 3939, 4578, 5265, 6000, 6783, 7614, 8493, 9420, 10395, 11418, 12489, 13608, 14775, 15990, 17253, 18564, 19923, 21330, 22785, 24288, 25839, 27438, 29085, 30780, 32523, 34314, 36153, 38040, 39975, 41958, 43989

Offset: 1

Emeric Deutsch, Sep 30 2010

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

			a(1)=15 because in D(5,1)=C_5 we have 5 distances equal to 1 and 5 distances equal to 2.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014642, A033991, A139273, A180578, A180867.

Maple
```
seq(3*n*(8*n-3), n = 1 .. 40);
```

Table[3n(8n-3),{n,40}] (* or *) LinearRecurrence[{3,-3,1},{15,78,189},40] (* Harvey P. Dale, May 01 2023 *)

a(n)=3*n*(8*n-3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n*(8*n-3).
a(n) = A180867(4,n).
The Wiener polynomial of the graph D(5,n) is nt(t+1)[2(n-1)t^2+2(n-1)t+5].
G.f.: -3*x*(11*x+5)/(x-1)^3. - Colin Barker, Oct 31 2012
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 3*exp(x)*x*(5 + 8*x).
a(n) = 3*A139273(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).

Original entry on oeis.org

3, 8, 14, 15, 40, 33, 27, 78, 96, 60, 42, 144, 189, 176, 95, 64, 228, 351, 348, 280, 138, 90, 352, 558, 648, 555, 408, 189, 125, 500, 864, 1032, 1035, 810, 560, 248, 165, 700, 1230, 1600, 1650, 1512, 1113, 736, 315, 216, 930, 1725, 2280, 2560, 2412, 2079, 1464

Offset: 3

Emeric Deutsch, Sep 30 2010

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

			Square array starts:
   3,  14,  33,  60,   95, ...
   8,  40,  96, 176,  280, ...
  15,  78, 189, 348,  555, ...
  27, 144, 351, 648, 1035, ...

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.

Cf. A014642, A033991, A180578, A180579.

T := proc (m, n) if `mod`(m, 2) = 1 then (1/8)*n*(m^2-1)*(2*m*n-m-2*n+2) else (1/8)*n*m^2*(2*m*n-m-2*n+2) end if end proc: for n from 3 to 12 do seq(T(n+1-j, j), j = 1 .. n-2) end do; # yields sequence in triangular form

T(3,n) = A033991(n).
T(4,n) = A014642(n).
T(5,n) = A180579(n).
T(6,n) = A180578(n).
T(m,n) = (1/8)n(m^2-1)(2mn-m-2n+2) if m is odd.
T(m,n) = (1/8)n*m^2*(2mn-m-2n+2) if m is even.
The Wiener polynomial of D(m,n) is n*w(m)+(1/2)n(n-1)r(m)^2, where, denoting S(t,p) = Sum_{j=1..p-1}t^j, we have w(m) = mS(t,m/2) + (1/2)mt^(m/2), r(m) = 2S(t,m/2) + t^(m/2) if m is even and w(m) = mS(t,(m+1)/2), r(m) = 2S(t,(m+1)/2) if m is odd.

Typos in Wiener polynomial information corrected by Emeric Deutsch, Sep 30 2010

A270704 Even 14-gonal (or tetradecagonal) numbers.

Original entry on oeis.org

0, 14, 76, 186, 344, 550, 804, 1106, 1456, 1854, 2300, 2794, 3336, 3926, 4564, 5250, 5984, 6766, 7596, 8474, 9400, 10374, 11396, 12466, 13584, 14750, 15964, 17226, 18536, 19894, 21300, 22754, 24256, 25806, 27404, 29050, 30744, 32486, 34276, 36114, 38000

Offset: 0

Ilya Gutkovskiy, Mar 22 2016

First bisection of A051866.

More generally, the ordinary generating function for the even k-gonal numbers with even k or for the first bisection of k-gonal numbers, is (k*x + (3*k - 8)*x^2)/(1 - x )^3.

OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

Cf. similar sequences of the even k-gonal numbers with even k: A016742 (k = 4), A014635 (k = 6), A014642 (k = 8), A028994 (k = 10), A193872 (k = 12).

LinearRecurrence[{3, -3, 1}, {0, 14, 76}, 41]
Table[2 n (12 n - 5), {n, 0, 40}]
PolygonalNumber[14,Range[0,80,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2017 *)

concat(0, Vec(2*x*(7 + 17*x)/(1 - x)^3 + O(x^60))) \\ Michel Marcus, Mar 22 2016

G.f.: 2*x*(7 + 17*x)/(1 - x)^3.
E.g.f.: 2*exp(x)*x*(7 + 12*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*n*(12*n - 5).
a(n) = A005843(n)*A017605(n-1).
Sum_{n>=1} 1/a(n) = (Pi - sqrt(3)*Pi + sqrt(3)*log(27) + sqrt(3)*log(64) + log(1728) + 6*log(sqrt(3)-1) + 2*sqrt(3)*log(sqrt(3)-1) - 6*log(sqrt(3)+1) - 2*sqrt(3)*log(sqrt(3)+1))/(20 + 20*sqrt(3)) = 0.102542837854…

cp's OEIS Frontend

A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).

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A180579 The Wiener index of the Dutch windmill graph D(5,n) (n>=1).

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A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A270704 Even 14-gonal (or tetradecagonal) numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula