A245828 -id:A245828 - cp's OEIS frontend

A245826 Triangle read by rows: T(m,n) is the Szeged index of the grid graph P_m X P_n (1 <= n <= m).

0, 1, 16, 4, 59, 216, 10, 144, 526, 1280, 20, 285, 1040, 2530, 5000, 35, 496, 1809, 4400, 8695, 15120, 56, 791, 2884, 7014, 13860, 24101, 38416, 84, 1184, 4316, 10496, 20740, 36064, 57484, 86016, 120, 1689, 6156, 14970, 29580, 51435, 81984, 122676, 174960, 165, 2320, 8455, 20560, 40625, 70640, 112595, 168480, 240285, 330000

Offset: 1

Emeric Deutsch, Aug 06 2014

T(n,1) = Szeged index of the path tree P_n = A000292(n-1).
T(n,2) = Szeged index of the ladder graph P_2 X P_n = A063521(n).
T(n,3) = Szeged index of the grid graph P_3 X P_n = A245827(n).
T(n,n) = Szeged index of the grid graph P_n X P_n = A245828(n).

			T(2,2) = 16 because P_2 X P_2 is the square C_4 and each of its 4 edges contributes 2*2=4 to its Szeged index.
Triangle starts:
0;
1,16;
4,59,216;
10,144,526,1280;
20,285,1040,2530,5000;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett., 9, 1996, 45-49.

Cf. A000292, A063521, A245827, A245828.

Cf. A245940 (row sums), A245941 (central terms).

a245826 n k = n * k * (2 * n^2 * k^2 - n^2 - k^2) `div` 6
a245826_row n = map (a245826 n) [1..n]
a245826_tabl = map a245826_row [1..]
-- Reinhard Zumkeller, Aug 07 2014

T:=proc(m,n) options operator, arrow: (1/6)*m*n*(2*m^2*n^2-m^2-n^2) end proc: for m to 12 do seq(T(m, n), n = 1 .. m) end do; # yields sequence in triangular form

T[m_, n_] := (1/6)*m*n*(2*m^2*n^2 - m^2 - n^2); Table[T[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Feb 09 2018 *)

T(m,n) = mn(2m^2 n^2 - m^2 - n^2)/6. See the Klavzar et al. reference; p. 47, line 6; there is a typo: n^2 - m^2 should be n^2 + m^2.

A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.

Original entry on oeis.org

0, 8, 72, 320, 1000, 2520, 5488, 10752, 19440, 33000, 53240, 82368, 123032, 178360, 252000, 348160, 471648, 627912, 823080, 1064000, 1358280, 1714328, 2141392, 2649600, 3250000, 3954600, 4776408, 5729472, 6828920, 8091000, 9533120, 11173888, 13033152, 15132040

Offset: 1

Emeric Deutsch, Sep 20 2008

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

			a(2)=8 because in P_2 x P_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2 (4*1 + 2*2 = 8).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (corrected by Ray Chandler, Jan 19 2019)
Dragan Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., Vol. 235, No. 1-3 (2001), pp. 237-244.
B.-Y. Yang and Y.-N. Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, Vol. 99 (2004), pp. 80-91.
Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., Vol. 135, No. 1-3 (1994), pp. 359-365.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Main diagonal of A143368.

Cf. A006414, A143944, A245828, A192828.

[n^3*(n^2-1)/3: n in [1..40]]; // Vincenzo Librandi, Feb 08 2014

Maple
```
seq((1/3)*n^3*(n^2-1),n=1..33);
```

Table[n^3 (n^2 - 1)/3, {n, 40}] (* Harvey P. Dale, Feb 07 2014 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 8, 72, 320, 1000, 2520}, 30] (* Harvey P. Dale, Feb 07 2014 *)
CoefficientList[Series[8 x (1 + 3 x + x^2)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 08 2014 *)

a(n)=n^3*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = Sum_{k=1..2n-2} k*A143944(n,k).
a(n) = n^3*(n^2-1)/3.
a(n) = 8*A006414(n-2). G.f.: 8*x^2*(1+3*x+x^2)/(x-1)^6. - R. J. Mathar, Sep 15 2010
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(2)=8, a(3)=72, a(4)=320, a(5)=1000, a(6)=2520, a(7)=5488. - Harvey P. Dale, Feb 07 2014
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=2} 1/a(n) = 15/4 - 3*zeta(3).
Sum_{n>=2} (-1)^n/a(n) = 9*zeta(3)/4 + 6*log(2) - 27/4. (End)

A245827 Szeged index of the grid graph P_3 X P_n.

Original entry on oeis.org

4, 59, 216, 526, 1040, 1809, 2884, 4316, 6156, 8455, 11264, 14634, 18616, 23261, 28620, 34744, 41684, 49491, 58216, 67910, 78624, 90409, 103316, 117396, 132700, 149279, 167184, 186466, 207176, 229365, 253084, 278384, 305316, 333931, 364280, 396414, 430384, 466241, 504036, 543820

Offset: 1

Emeric Deutsch, Aug 06 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett., 9, 1996, 45-49.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A245826, A063521, A245828.

[(1/2)*n*(17*n^2 - 9): n in [1..40]]; // Vincenzo Librandi, Aug 07 2014

a := proc (n) options operator, arrow: (1/2)*n*(17*n^2-9) end proc: seq(a(n), n = 1 .. 40);

CoefficientList[Series[(4 x^2 + 43 x + 4)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *)
LinearRecurrence[{4,-6,4,-1},{4,59,216,526},40] (* Harvey P. Dale, Oct 21 2017 *)

Vec(x*(4*x^2+43*x+4)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 07 2014

a(n) = (1/2)*n*(17*n^2 - 9).
a(n) = A245826(n, 3).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: x*(4*x^2+43*x+4) / (x-1)^4. - Colin Barker, Aug 07 2014

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A245826 Triangle read by rows: T(m,n) is the Szeged index of the grid graph P_m X P_n (1 <= n <= m).

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A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.

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A245827 Szeged index of the grid graph P_3 X P_n.

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