A180863 Wiener index of the n-sun graph.

6, 21, 44, 75, 114, 161, 216, 279, 350, 429, 516, 611, 714, 825, 944, 1071, 1206, 1349, 1500, 1659, 1826, 2001, 2184, 2375, 2574, 2781, 2996, 3219, 3450, 3689, 3936, 4191, 4454, 4725, 5004, 5291, 5586, 5889, 6200, 6519, 6846, 7181, 7524, 7875, 8234, 8601

Offset: 2

Author

Emeric Deutsch, Sep 28 2010

Keywords

• nonn
• easy

Comments

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

The Wiener polynomial of the n-sun graph is (1/2)*nt[(n-3)t^2+2(n-1)t+n+3].

Comment from Zachary Dove, Apr 19 2021 (Start)

(Cf. A343560)

On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie parallel to the positive x-axis, located within the first quadrant, as seen in the example below:

.

99 64--65--66--67--68--69--70--71--72

| | |

98 63 36--37--38--39--40--41--42 73

| | | | |

97 62 35 16--17--18--19--20 43 74

| | | | | | |

96 61 34 15 4---5--*6**21**44**75*

| | | | | | | | |

95 60 33 14 3 0 7 22 45 76

| | | | | | | | | |

94 59 32 13 2---1 8 23 46 77

| | | | | | | |

93 58 31 12--11--10---9 24 47 78

| | | | | |

92 57 30--29--28--27--26--25 48 79

| | | |

91 56--55--54--53--52--51--50--49 80

| |

90--89--88--87--86--85--84--83--82--81

(End)

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.
• Eric Weisstein's World of Mathematics, Sun Graph.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Maple

seq(n*(4*n-5), n = 2 .. 50);

Mathematica

(* Start from Eric W. Weisstein, Sep 07 2017, adapted to new offset *)
Table[n (4 n - 5), {n, 2, 20}]
LinearRecurrence[{3, -3, 1}, {6, 21, 44}, 20]
CoefficientList[Series[(-6 - 3 x + x^2)/(-1 + x)^3, {x, 0, 20}], x]
(* End *)

PARI

a(n)=n*(4*n-5) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = n*(4*n-5).

G.f.: x^2*(-6-3*x+x^2)/(x-1)^3. - Colin Barker, Oct 31 2012, adapted to new offset Sep 29 2021

a(n) = 3*a(n-1) - 3*a(n-2) + a(n). - Eric W. Weisstein, Sep 07 2017

a(n) = A033954(n-1)-1 = A033951(n-1) -1. -R. J. Mathar, Sep 29 2021

Sum_{n>=2} 1/a(n) = 1/5 - Pi/10 + 3*log(2)/5. - Amiram Eldar, Apr 16 2022

E.g.f.: exp(x)*(-x + 4*x^2) + x. - Nikolaos Pantelidis, Feb 10 2023

Extensions

a(2)=6 prefixed by R. J. Mathar, Sep 29 2021