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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A180859 Square array read by antidiagonals: T(m,n) is the Wiener index of the windmill graph D(m,n) obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs; m>=2, n>=1).

Original entry on oeis.org

1, 3, 4, 6, 14, 9, 10, 30, 33, 16, 15, 52, 72, 60, 25, 21, 80, 126, 132, 95, 36, 28, 114, 195, 232, 210, 138, 49, 36, 154, 279, 360, 370, 306, 189, 64, 45, 200, 378, 516, 575, 540, 420, 248, 81, 55, 252, 492, 700, 825, 840, 742, 552, 315, 100, 66, 310, 621, 912, 1120, 1206, 1155, 976, 702, 390, 121
Offset: 2

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Author

Emeric Deutsch, Sep 25 2010

Keywords

Comments

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of nodes in the graph.
For the Wiener indices of D(3,n), D(4,n), D(5,n) and D(6,n) see A033991, A152743, A028994, and A180577, respectively.

Examples

			T(3,2)=14 because the graph D(3,2) consists of two triangles OAB and OCD with a common node O; it has 6 distances equal to 1 (the edges) and 4 distances equal to 2 (AC, AD, BC, and BD); 6 * 1 + 4 * 2 = 14.
Square array starts:
   1,   4,   9,  16,  25, ...
   3,  14,  33,  60,  95, ...
   6,  30,  72, 132, 210, ...
  10,  52, 126, 232, 370, ...

Links

Crossrefs

Cf. A028994, A033991, A152743, A180577.

Programs

  • Maple
    T := proc (m, n) options operator, arrow: (1/2)*n*(m-1)*((m-1)*(2*n-1)+1) end proc: for p from 2 to 12 do seq(T(p+1-j, j), j = 1 .. p-1) end do; # yields sequence in triangular form
  • PARI
    T(m,n) = (1/2)*n*(m-1)*((m-1)*(2*n-1)+1);
antidiag(n) = vector(n-1, k, k; T(n-k+1, k)); \\ Michel Marcus, Mar 09 2023

Formula

T(m,n) = (1/2)n(m-1)((m-1)(2n-1)+1).
The Wiener polynomial of D(m,n) is (1/2)n(m-1)t((m-1)(n-1)t+m).

A268579 Expansion of (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).

Original entry on oeis.org

1, 7, 11, 41, 48, 120, 130, 262, 275, 485, 501, 807, 826, 1246, 1268, 1820, 1845, 2547, 2575, 3445, 3476, 4532, 4566, 5826, 5863, 7345, 7385, 9107, 9150, 11130, 11176, 13432, 13481, 16031, 16083, 18945, 19000, 22192, 22250, 25790, 25851, 29757, 29821
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 21 2016

Keywords

Examples

			a(0) = 1;
a(1) = 1 + 2*3 = 7;
a(2) = 1 + 2*3 + 4 = 11;
a(3) = 1 + 2*3 + 4 + 5*6 = 41;
a(4) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(5) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(6) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12= 262;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 = 275;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 + 14*15 = 485, etc.

Links

Crossrefs

Cf. A000027, A001651, A008585, A016777, A135036, A152743.

Programs

  • Mathematica
    Table[Sum[(6 k + (-1)^k + 3) ((3 k - (-1)^k (3 k + 1) + 5)/16), {k, 0, n}], {n, 0, 42}]
Table[1 + (n (6 n^2 + 27 n + 35) - (9 n^2 + 15 n + 2) (-1)^n + 2)/16, {n, 0, 42}]
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 7, 11, 41, 48, 120, 130}, 43]
  • PARI
    Vec((1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3) + O(x^50)) \\ Michel Marcus, Feb 21 2016

Formula

G.f.: (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).
a(n) = Sum_{k = 0..n} (6*k + (-1)^k +3)*(3*k - (-1)^k*(3*k + 1) + 5)/16.
a(n) = 1 + (n*(6*n^2 + 27*n + 35) - (9*n^2 + 15*n + 2)*(-1)^n + 2)/16.
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