A143942 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n squares joined at vertices (i.e., joined like <><><>...<>; here <> is a square!); 1 <= k <= 2n.

4, 2, 8, 8, 4, 1, 12, 14, 8, 6, 4, 1, 16, 20, 12, 11, 8, 6, 4, 1, 20, 26, 16, 16, 12, 11, 8, 6, 4, 1, 24, 32, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 28, 38, 24, 26, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 32, 44, 28, 31, 24, 26, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 36, 50, 32, 36, 28, 31, 24, 26

Offset: 1

Author

Emeric Deutsch, Sep 06 2008

Keywords

• nonn
• tabf

Comments

Row n has 2n entries.

The entries in row n are the coefficients of the Wiener polynomial of the linear chain of n squares.

Sum of entries in row n = 3n(3n+1)/2 = A081266(n).

Sum_{k=1..n} k*T(n,k) = the Wiener index of a linear chain of n squares joined at vertices (like <><><>...) = A143943(n).

Examples

			T(2,1)=8 because the chain of 2 squares (<><>) has 8 edges.
Triangle starts:
   4,  2;
   8,  8,  4,  1;
  12, 14,  8,  6,  4,  1;
  16, 20, 12, 11,  8,  6,  4,  1;
  20, 26, 16, 16, 12, 11,  8,  6,  4,  1;
		

Links

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Crossrefs

Cf. A081266, A143943.

Programs

Maple

T:=proc(n,k) if 2*n < k then 0 elif k = 1 then 4*n elif k = 2 then 6*n-4 elif `mod`(k,2)=1 then 4*n-2*k+2 elif `mod`(k,2)=0 then 5*n-(5/2)*k+1 else 0 end if end proc: for n to 10 do seq(T(n,k),k=1..2*n) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := Which[2n < k, 0, k == 1, 4n, k == 2, 6n - 4, OddQ[k], 4n - 2k + 2, EvenQ[k], 5n - (5/2) k + 1, True, 0];
Table[T[n, k], {n, 1, 10}, {k, 1, 2n}] // Flatten (* Jean-François Alcover, Aug 23 2024, after Maple program *)

Formula

T(n,1) = 4n; T(n,2) = 6n-4; T(n,2p+1) = 4(n-p); T(n,2p) = 5(n-p)+1.

G.f. = G(q,z) = qz/(4+2q+4qz-q^3*z)/((1-q^2*z)*(1-z)^2).