a(n) = (1/8)*n*(n^2 +2n 2*n + 8) if n is even; a(n) = (1/8)*(n^3 +2n 2*n^2 +7n 7*n - 2) if n is odd.

From R. J. Mathar, Sep 29 2010: (Start)

a(n) = +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6). - _R. J. Mathar_, Sep 29 2010

G.f.: (x ^3*(1 + 2 *x - x^2 + x^4))/((-1 + - x)^4 *(1 + x)^2). - _R. J. Mathar_, Sep 29 2010(End)

a(n) = (-2 + 15*n + 4*n^2 + 2*n^3 + (-1)^n* (2 + n))/16. - 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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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.

a(n)=A180860(n,1).

a(n) = A180860(n,1).

a(n) = (1/8)n(n^2+2n+8) if n is even; a(n)=(1/8)(n^3+2n^2+7n-2) if n is odd.

a(n)=(1/8)(n^3+2n^2+7n-2) if n is odd.

Cf. A180860

First column of A180860.

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The Wiener index of the n-pan graph. 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.

Wiener index of the n-pan graph.

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

3,12

Extended to a(1)-a(2) using the formula/recurrence.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TadpoleGraph.html">Tadpole Graph</a>.

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

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^3*(8-14*x^2+6*x^3+7*x^4-4*x^5)/((1+x)^2*(x-1)^4). [From _ _R. J. Mathar_, Sep 29 2010]

G.f.: (x (1 + 2 x - x^2 + x^4))/((-1 + x)^4 (1 + x)^2). - R. J. Mathar, Sep 29 2010

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 = 3 1 .. 50);

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 4, 8, 16, 26, 42, 61, 88}, 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 *)

a(1)-a(2) from Eric W. Weisstein, Sep 08 2017

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<a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1). [From _R. J. Mathar_, Sep 29 2010]

nonn,easy

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<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1). [From R. J. Mathar, Sep 29 2010]

<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1). [From R. J. Mathar, Sep 29 2010]

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