OFFSET
2,1
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).
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
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
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 *)
PROG
(PARI) a(n)=n*(4*n-5) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Sep 28 2010
EXTENSIONS
a(2)=6 prefixed by R. J. Mathar, Sep 29 2021
STATUS
approved