# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a180863
Showing 1-1 of 1

%I A180863 #33 Feb 11 2023 16:21:11
%S A180863 6,21,44,75,114,161,216,279,350,429,516,611,714,825,944,1071,1206,
%T A180863 1349,1500,1659,1826,2001,2184,2375,2574,2781,2996,3219,3450,3689,
%U A180863 3936,4191,4454,4725,5004,5291,5586,5889,6200,6519,6846,7181,7524,7875,8234,8601
%N A180863 Wiener index of the n-sun graph.
%C A180863 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%C A180863 The Wiener polynomial of the n-sun graph is (1/2)*nt[(n-3)t^2+2(n-1)t+n+3].
%C A180863 Comment from _Zachary Dove_, Apr 19 2021 (Start)
%C A180863    (Cf. A343560)
%C A180863 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 A180863 .
%C A180863   99  64--65--66--67--68--69--70--71--72
%C A180863    |   |                               |
%C A180863   98  63  36--37--38--39--40--41--42  73
%C A180863    |   |   |                       |   |
%C A180863   97  62  35  16--17--18--19--20  43  74
%C A180863    |   |   |   |               |   |   |
%C A180863   96  61  34  15   4---5--*6**21**44**75*
%C A180863    |   |   |   |   |       |   |   |   |
%C A180863   95  60  33  14   3   0   7  22  45  76
%C A180863    |   |   |   |   |   |   |   |   |   |
%C A180863   94  59  32  13   2---1   8  23  46  77
%C A180863    |   |   |   |           |   |   |   |
%C A180863   93  58  31  12--11--10---9  24  47  78
%C A180863    |   |   |                   |   |   |
%C A180863   92  57  30--29--28--27--26--25  48  79
%C A180863    |   |                           |   |
%C A180863   91  56--55--54--53--52--51--50--49  80
%C A180863    |                                   |
%C A180863   90--89--88--87--86--85--84--83--82--81
%C A180863 (End)
%H A180863 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 A180863 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>.
%H A180863 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A180863 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A180863 a(n) = n*(4*n-5).
%F A180863 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 A180863 a(n) = 3*a(n-1) - 3*a(n-2) + a(n). - _Eric W. Weisstein_, Sep 07 2017
%F A180863 a(n) = A033954(n-1)-1 = A033951(n-1) -1. -_R. J. Mathar_, Sep 29 2021
%F A180863 Sum_{n>=2} 1/a(n) = 1/5 - Pi/10 + 3*log(2)/5. - _Amiram Eldar_, Apr 16 2022
%F A180863 E.g.f.: exp(x)*(-x + 4*x^2) + x. - _Nikolaos Pantelidis_, Feb 10 2023
%p A180863 seq(n*(4*n-5), n = 2 .. 50);
%t A180863 (* Start from _Eric W. Weisstein_, Sep 07 2017, adapted to new offset *)
%t A180863 Table[n (4 n - 5), {n, 2, 20}]
%t A180863 LinearRecurrence[{3, -3, 1}, {6, 21, 44}, 20]
%t A180863 CoefficientList[Series[(-6 - 3 x + x^2)/(-1 + x)^3, {x, 0, 20}], x]
%t A180863 (* End *)
%o A180863 (PARI) a(n)=n*(4*n-5) \\ _Charles R Greathouse IV_, Jun 17 2017
%K A180863 nonn,easy
%O A180863 2,1
%A A180863 _Emeric Deutsch_, Sep 28 2010
%E A180863 a(2)=6 prefixed by _R. J. Mathar_, Sep 29 2021

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE