A143943 -id:A143943 - cp's OEIS frontend

A304160 a(n) = n^4 - 3n^3 + 6n^2 - 5*n + 2 (n >= 1).

1, 8, 41, 142, 377, 836, 1633, 2906, 4817, 7552, 11321, 16358, 22921, 31292, 41777, 54706, 70433, 89336, 111817, 138302, 169241, 205108, 246401, 293642, 347377, 408176, 476633, 553366, 639017, 734252, 839761, 956258, 1084481, 1225192, 1379177, 1547246, 1730233, 1928996, 2144417, 2377402

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the second Zagreb index of the Barbell graph B(n) (n>=3).
The Barbell graph B(n) is defined as two copies of the complete graph K(n) (n>=3), connected by a bridge.
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)*d(j) over all edges ij of the graph.
The M-polynomial of the Barbell graph B(n) is M(B(n),x,y) = (n-1)*(n-2)*x^{n-1}*y^{n-1} + 2*(n-1)*x^{n-1}*y^n + x^n*y^n.

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavžar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Eric Weisstein's World of Mathematics, Independent Edge Set

Cf. A000583, A143943.

Vec(x*(1 + 3*x + 11*x^2 + 7*x^3 + 2*x^4) / (1 - x)^5 + O(x^60)) \\ Colin Barker, May 09 2018

From Colin Barker, May 09 2018: (Start)
G.f.: x*(1 + 3*x + 11*x^2 + 7*x^3 + 2*x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)
a(n) = A000583(n) - A143943(n-1), assuming that A143943(0) = 0. - Omar E. Pol, May 09 2018

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.

Original entry on oeis.org

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

Emeric Deutsch, Sep 06 2008

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).

			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;

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

Cf. A081266, A143943.

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

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 *)

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).

cp's OEIS Frontend

A304160 a(n) = n^4 - 3n^3 + 6n^2 - 5*n + 2 (n >= 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A304160 a(n) = n^4 - 3*n^3 + 6*n^2 - 5*n + 2 (n >= 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A304160 a(n) = n^4 - 3n^3 + 6n^2 - 5*n + 2 (n >= 1).