A143945 -id:A143945 - cp's OEIS frontend

A006414 Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

1, 9, 40, 125, 315, 686, 1344, 2430, 4125, 6655, 10296, 15379, 22295, 31500, 43520, 58956, 78489, 102885, 133000, 169785, 214291, 267674, 331200, 406250, 494325, 597051, 716184, 853615, 1011375, 1191640, 1396736, 1629144, 1891505, 2186625, 2517480, 2887221

Offset: 0

N. J. A. Sloane

The number of faces is 1.
a(n) = K(Oa(2,3,n)), Kekulé numbers of certain benzenoid structures (see the Cyvin - Gutman reference).
Sequence of partial sums of A006322. - L. Edson Jeffery, Dec 13 2011
The sequence b(n) = a(n-2) with a(-1) = 0, for n >= 1, is b(n) = n^3*(n^2 - 1)/4!. It is obtained by comparing the result for the powers n^5 from Worpitzky's identity (see a formula in A000584) with the result obtained from the counting of degrees of freedom for the decomposition of a rank 5 tensor in n dimensions via the standard Young tableaux version with 5 boxes corresponding to the seven partitions of 5. The difference of the two versions gives: 10*(binomial(n+3, 5) + 3*binomial(n+2, 5) + binomial(n+1, 5)) = 5*n*(binomial(n+2, 4) + binomial(n+1, 4)) = 10*b(n). See the formula for a(n) below. - Wolfdieter Lang, Jul 18 2019

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988, p. 105, eq. (ii), and p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Differences of A006542 (C(n, 3)*C(n-1, 3)/4).

Cf. A004068, A005891, A006322, A006415, A006434, A006441, A133754, A143945.

List([0..40], n-> (n+2)^2*Binomial(n+3,3)/4 ); G. C. Greubel, Sep 02 2019

[(n+1)*(n+2)^3*(n+3)/24: n in [0..40]]; // Wesley Ivan Hurt, May 10 2014

seq((n+2)^2*binomial(n+3,3)/4, n=0..40); # G. C. Greubel, Sep 02 2019

Table[(n + 1)*(n + 2)^3*(n + 3)/24, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = (n+1)*(n+2)^3*(n+3)/24; \\ Michel Marcus, Jul 09 2017

[(n+2)^2*binomial(n+3,3)/4 for n in (0..40)] # G. C. Greubel, Sep 02 2019

a(n) = (n+1)*(n+2)^3*(n+3)/24. - N. J. A. Sloane, Apr 02 2004
a(n) = (n+2)^3*((n+2)^2 - 1)/24. - Paul Richards, Mar 04 2007
G.f.: (1 + 3*x + x^2)/(1-x)^6. - Colin Barker, Feb 21 2012
a(n) = (Sum_{k=0..n+1} k*(n+1)*((n+1)^2 - k^2))/6 for n > 0, which is the sum of all areas of Pythagorean triangles with arms 2*k*(n+1) and (n+1)^2 - k^2 with hypotenuse k^2 + (n+1)^2. - J. M. Bergot, May 12 2014
a(n) = A143945(n+2)/8. - J. M. Bergot, Jun 14 2014
Sum_{n>=0} 1/a(n) = 30 - 24*zeta(3). - Jaume Oliver Lafont, Jul 09 2017
a(n) = binomial(n+5, 5) + 3*binomial(n+4, 5) + binomial(n+3, 5) = ((n+2)/2)*(binomial(n+4, 4) + binomial(n+3, 4)), for n >= 0. See a comment above on the sequence b(n) = a(n-2) = n^3*(n^2 - 1)/4!. - Wolfdieter Lang, Jul 19 2019
E.g.f.: (24 + 192*x + 276*x^2 + 124*x^3 + 20*x^4 + x^5)*exp(x)/4!. - G. C. Greubel, Sep 02 2019
Sum_{n>=0} (-1)^n/a(n) = 18*zeta(3) + 48*log(2) - 54. - Amiram Eldar, Jan 09 2022

More terms from Robert Newstedt (Patternfinder(AT)webtv.net)

Name clarified by Andrew Howroyd, Apr 05 2021

A143368 Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).

Original entry on oeis.org

0, 1, 8, 4, 25, 72, 10, 56, 154, 320, 20, 105, 280, 570, 1000, 35, 176, 459, 920, 1595, 2520, 56, 273, 700, 1386, 2380, 3731, 5488, 84, 400, 1012, 1984, 3380, 5264, 7700, 10752, 120, 561, 1404, 2730, 4620, 7155, 10416, 14484, 19440

Offset: 1

Emeric Deutsch, Sep 05 2008

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

This is the lower triangular half of a symmetric square array.

			Presentation as symmetric square array starts:
======================================================
n\k|   1   2    3    4    5    6     7     8     9
---|--------------------------------------------------
1  |   0   1    4   10   20   35    56    84   120 ...
2  |   1   8   25   56  105  176   273   400   561 ...
3  |   4  25   72  154  280  459   700  1012  1404 ...
4  |  10  56  154  320  570  920  1386  1984  2730 ...
5  |  20 105  280  570 1000 1595  2380  3380  4620 ...
6  |  35 176  459  920 1595 2520  3731  5264  7155 ...
7  |  56 273  700 1386 2380 3731  5488  7700 10416 ...
8  |  84 400 1012 1984 3380 5264  7700 10752 14484 ...
9  | 120 561 1404 2730 4620 7155 10416 14484 19440 ...
... - _Andrew Howroyd_, May 27 2017
T(2,2)=8 because in a square we have four distances equal to 1 and two distances equal to 2.
T(2,1)=1 because on the path graph on two vertices there is one distance equal to 1.
T(3,2)=25 because on the P(2) X P(3) graph there are 7 distances equal to 1, 6 distances equal to 2 and 2 distances equal to 3, with 7*1 + 6*2 + 2*3 = 25.
Triangle starts: 0; 1,8; 4,25,72; 10,56,154,320;

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem., 8 (1991), 53-62.
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, Grid Graph
Eric Weisstein's World of Mathematics, Wiener Index

Cf. A180569 (row 3), A131423 (row 2).
Main diagonal is A143945.
Cf. A245826.

T:=proc(n, k) options operator, arrow: (1/6)*k*n*(n+k)*(k*n-1) end proc: for n to 9 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, May 28 2017 *)

T(n,k)=k*n*(n+k)*(k*n-1)/6;
for (n=1,8,for(k=1,8,print1(T(n,k),", "));print) \\ Andrew Howroyd, May 27 2017

T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).

A143944 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k from each other in the grid P_n X P_n (1 <= k <= 2n-2), where P_n is the path graph on n vertices.

Original entry on oeis.org

4, 2, 12, 14, 8, 2, 24, 34, 32, 20, 8, 2, 40, 62, 68, 60, 40, 20, 8, 2, 60, 98, 116, 116, 100, 70, 40, 20, 8, 2, 84, 142, 176, 188, 180, 154, 112, 70, 40, 20, 8, 2, 112, 194, 248, 276, 280, 262, 224, 168, 112, 70, 40, 20, 8, 2, 144, 254, 332, 380, 400, 394, 364, 312, 240

Offset: 2

Emeric Deutsch, Sep 19 2008

Row n contains 2n-2 entries.
Sum of entries in row n = n^2*(n^2 - 1)/2 = A083374(n).
The entries in row n are the coefficients of the Wiener (Hosoya) polynomial of the grid P_n X P_n.
Sum_{k=1..2n-2} k*T(n,k) = n^3*(n^2 - 1)/3 = A143945(n) = the Wiener index of the grid P_n X P_n.
The average of all distances in the grid P_n X P_n is 2n/3.

			T(2,2)=2 because P_2 X P_2 is a square and there are 2 pairs of vertices at distance 2.
Triangle starts:
   4,  2;
  12, 14,  8,  2;
  24, 34, 32, 20,  8,  2;
  40, 62, 68, 60, 40, 20,  8,  2;

D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.
B.-Y. Yang and Y.-N. Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91.
Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., 135 (1994), 359-365.

Cf. A083374, A143945.

for n from 2 to 10 do Q[n]:=sort(expand(simplify((1/2)*(2*q*(1-q^n)-n*(1-q^2))^2/(1-q)^4-(1/2)*n^2))) end do: for n from 2 to 9 do seq(coeff(Q[n],q,j),j= 1..2*n-2) end do;

Generating polynomial of row n is (2q(1-q^n) - n(1-q^2))^2/(2(1-q)^4) - n^2/2.

A302298 Wiener index of the graph of nodes (i,j) of the square lattice such that abs(i) + abs(j) <= n.

Original entry on oeis.org

0, 16, 192, 1008, 3504, 9504, 21840, 44576, 83232, 145008, 239008, 376464, 570960, 838656, 1198512, 1672512, 2285888, 3067344, 4049280, 5268016, 6764016, 8582112, 10771728, 13387104, 16487520, 20137520, 24407136, 29372112, 35114128, 41721024, 49287024, 57912960, 67706496

Offset: 0

Andres Cicuttin, Apr 04 2018

nonn

The considered grid distance is the Manhattan distance (taxicab metric).

Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Wiener Index
Eric Weisstein's World of Mathematics, Taxicab Metric

Cf. A292053, A143945.

a[n_]:=(1/2)*Sum[Sum[Sum[Sum[
Abs[i2-i1] + Abs[j2-j1],
{j1,Abs[i1]-n,n-Abs[i1]}],{i1,-n,n}],
{j2,Abs[i2]-n,n-Abs[i2]}],{i2,-n,n}];
Table[a[n],{n,0,32}]

Conjectures from Colin Barker, Apr 08 2018: (Start)
G.f.: 16*x*(1 + x)*(1 + 5*x + x^2) / (1 - x)^6.
a(n) = 2*(n*(6 + 25*n + 40*n^2 + 35*n^3 + 14*n^4)) / 15.
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) for n>5.
(End)

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A006414 Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

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A143368 Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).

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A143944 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k from each other in the grid P_n X P_n (1 <= k <= 2n-2), where P_n is the path graph on n vertices.

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Examples

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Crossrefs

Programs

Maple

Formula

A302298 Wiener index of the graph of nodes (i,j) of the square lattice such that abs(i) + abs(j) <= n.

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Author

Keywords

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Programs

Mathematica

Formula