A292044 Wiener index of the n-halved cube graph.

0, 1, 6, 32, 160, 768, 3584, 16384, 73728, 327680, 1441792, 6291456, 27262976, 117440512, 503316480, 2147483648, 9126805504, 38654705664, 163208757248, 687194767360, 2886218022912, 12094627905536, 50577534877696, 211106232532992, 879609302220800, 3659174697238528

Offset: 1

Author

Eric W. Weisstein, Sep 08 2017

Keywords

• nonn
• easy

Comments

a(n) is the sum of the first 2^(n-1) entries of A116640. - Joe Slater, Apr 11 2018

Links

• Eric Weisstein's World of Mathematics, Halved Cube Graph.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index entries for linear recurrences with constant coefficients, signature (8,-16).

Crossrefs

Cf. A116640.

Programs

Mathematica

Table[If[n == 1, 0, 2^(2 n - 5) n], {n, 40}]
Join[{0}, LinearRecurrence[{8, -16}, {1, 6}, 20]]
CoefficientList[Series[((1 - 2 x) x)/(1 - 4 x)^2, {x, 0, 20}], x]

PARI

a(n) = if(n<2, n-1, 2^(2*n-5)*n); \\ Altug Alkan, Apr 12 2018

Formula

a(n) = 2^(2*n-5)*n for n > 1.

a(n) = 8*a(n-1) - 16*a(n-2) for n > 3.

G.f.: ((1 - 2 x) x^2)/(1 - 4 x)^2.

a(n) = 4*a(n-1) + 2^(2*n-5) for n > 2. - Joe Slater, Apr 11 2018

From Amiram Eldar, Apr 16 2022: (Start)

Sum_{n>=2} 1/a(n) = 32*log(4/3) - 8.

Sum_{n>=2} (-1)^n/a(n) = 8 - 32*log(5/4). (End)

From Stefano Spezia, Aug 04 2022: (Start)

E.g.f.: (exp(4*x) - 1)*x/8.

a(n) = (-1)^n*det(M(n-1))/2 for n > 1, where M(n) is the n X n symmetric Toeplitz matrix whose first row consists of 2, 4, ..., 2*n. (End)