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A180861
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Wiener index of the n-pan graph.
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1
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1, 4, 8, 16, 26, 42, 61, 88, 119, 160, 206, 264, 328, 406, 491, 592, 701, 828, 964, 1120, 1286, 1474, 1673, 1896, 2131, 2392, 2666, 2968, 3284, 3630, 3991, 4384, 4793, 5236, 5696, 6192, 6706, 7258, 7829, 8440, 9071, 9744, 10438, 11176, 11936, 12742, 13571, 14448
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OFFSET
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1,2
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COMMENTS
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The n-pan graph is obtained by joining with an edge a node in the cycle graph C_n to the singleton graph P_1. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
The n-pan graph is a special case of the tadpole graph.
Extended to a(1)-a(2) using the formula/recurrence.
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LINKS
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Eric Weisstein's World of Mathematics, Pan Graph
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FORMULA
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a(n) = (1/8)*n*(n^2 + 2*n + 8) if n is even; a(n) = (1/8)*(n^3 + 2*n^2 + 7*n - 2) if n is odd.
a(n) = +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
G.f.: x*(1 + 2*x - x^2 + x^4)/((1 - x)^4*(1 + x)^2). (End)
a(n) = (-2 + 15*n + 4*n^2 + 2*n^3 + (-1)^n*(2 + n))/16. - Eric W. Weisstein, Sep 08 2017
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EXAMPLE
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a(3)=8 because the graph consists of a triangle ABCA and an edge AD; the distances are d(A,B)=d(B,C)=d(C,A)=d(A,D)=1 and d(DB)=d(DC)=2.
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MAPLE
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a := proc (n) if `mod`(n, 2) = 0 then (1/8)*n*(n^2+2*n+8) else (1/8)*n^3+(1/4)*n^2+(7/8)*n-1/4 end if end proc: seq(a(n), n = 1 .. 50);
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MATHEMATICA
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LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 4, 8, 16, 26, 42}, 50] (* Harvey P. Dale, Jun 04 2015 *)
Table[(-2 + 15 n + 4 n^2 + 2 n^3 + (-1)^n (2 + n))/16, {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[(1 + 2 x - x^2 + x^4)/((1 - x)^4 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)
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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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