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A180863
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Wiener index of the n-sun graph.
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2
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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
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OFFSET
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2,1
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COMMENTS
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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].
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:
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99 64--65--66--67--68--69--70--71--72
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98 63 36--37--38--39--40--41--42 73
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97 62 35 16--17--18--19--20 43 74
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96 61 34 15 4---5--*6**21**44**75*
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95 60 33 14 3 0 7 22 45 76
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94 59 32 13 2---1 8 23 46 77
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93 58 31 12--11--10---9 24 47 78
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92 57 30--29--28--27--26--25 48 79
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91 56--55--54--53--52--51--50--49 80
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90--89--88--87--86--85--84--83--82--81
(End)
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LINKS
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Eric Weisstein's World of Mathematics, Sun Graph.
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FORMULA
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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
Sum_{n>=2} 1/a(n) = 1/5 - Pi/10 + 3*log(2)/5. - Amiram Eldar, Apr 16 2022
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MAPLE
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seq(n*(4*n-5), n = 2 .. 50);
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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