A292044 - OEIS

%I #34 Aug 04 2022 15:51:01

%S 0,1,6,32,160,768,3584,16384,73728,327680,1441792,6291456,27262976,

%T 117440512,503316480,2147483648,9126805504,38654705664,163208757248,

%U 687194767360,2886218022912,12094627905536,50577534877696,211106232532992,879609302220800,3659174697238528

%N Wiener index of the n-halved cube graph.

%C a(n) is the sum of the first 2^(n-1) entries of A116640. - _Joe Slater_, Apr 11 2018

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HalvedCubeGraph.html">Halved Cube Graph</a>.

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).

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

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

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

%F a(n) = 4*a(n-1) + 2^(2*n-5) for n > 2. - _Joe Slater_, Apr 11 2018

%F From _Amiram Eldar_, Apr 16 2022: (Start)

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

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

%F From _Stefano Spezia_, Aug 04 2022: (Start)

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

%F 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)

%t Table[If[n == 1, 0, 2^(2 n - 5) n], {n, 40}]

%t Join[{0}, LinearRecurrence[{8, -16}, {1, 6}, 20]]

%t CoefficientList[Series[((1 - 2 x) x)/(1 - 4 x)^2, {x, 0, 20}], x]

%o (PARI) a(n) = if(n<2, n-1, 2^(2*n-5)*n); \\ _Altug Alkan_, Apr 12 2018

%Y Cf. A116640.

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Sep 08 2017