%I #33 Feb 11 2023 16:21:11

%S 6,21,44,75,114,161,216,279,350,429,516,611,714,825,944,1071,1206,

%T 1349,1500,1659,1826,2001,2184,2375,2574,2781,2996,3219,3450,3689,

%U 3936,4191,4454,4725,5004,5291,5586,5889,6200,6519,6846,7181,7524,7875,8234,8601

%N Wiener index of the n-sun graph.

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

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

%C Comment from _Zachary Dove_, Apr 19 2021 (Start)

%C (Cf. A343560)

%C 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:

%C .

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

%C | | |

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

%C | | | | |

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

%C | | | | | | |

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

%C | | | | | | | | |

%C 95 60 33 14 3 0 7 22 45 76

%C | | | | | | | | | |

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

%C | | | | | | | |

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

%C | | | | | |

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

%C | | | |

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

%C | |

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

%C (End)

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60 (1996), 959-969.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

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

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n). - _Eric W. Weisstein_, Sep 07 2017

%F a(n) = A033954(n-1)-1 = A033951(n-1) -1. -_R. J. Mathar_, Sep 29 2021

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

%F E.g.f.: exp(x)*(-x + 4*x^2) + x. - _Nikolaos Pantelidis_, Feb 10 2023

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

%t (* Start from _Eric W. Weisstein_, Sep 07 2017, adapted to new offset *)

%t Table[n (4 n - 5), {n, 2, 20}]

%t LinearRecurrence[{3, -3, 1}, {6, 21, 44}, 20]

%t CoefficientList[Series[(-6 - 3 x + x^2)/(-1 + x)^3, {x, 0, 20}], x]

%t (* End *)

%o (PARI) a(n)=n*(4*n-5) \\ _Charles R Greathouse IV_, Jun 17 2017

%K nonn,easy

%O 2,1

%A _Emeric Deutsch_, Sep 28 2010

%E a(2)=6 prefixed by _R. J. Mathar_, Sep 29 2021