A221047 The hyper-Wiener index of the Bethe cactus lattice graph E_n defined pictorially in the Hosoya - Balasubramanian reference.

5, 406, 11458, 221572, 3519703, 49623850, 646314724, 7958362600, 93998378377, 1075239211294, 11991495728998, 131012033254444, 1407240588517147, 14901371119404658, 155885324936843080, 1613748962415344464, 16554187503550561933, 168462466356459175462

Offset

1,1

Links

Ray Chandler, Table of n, a(n) for n = 1..1000

K. Balasubramanian, Recent developments in tree-pruning methods and polynomials for cactus graphs and trees, J. Math. Chemistry, 4 (1990) 89-102.

H. Hosoya and K. Balasubramanian, Exact dimer statistics and characteristic polynomials of cacti lattices, Theor. Chim. Acta 76 (1989) 315-329. Also on ResearchGate.

Index entries for linear recurrences with constant coefficients, signature (37, -549, 4185, -17523, 40095, -45927, 19683).

Formula

a(n) = -(1/8)+3^n*(2*n^2/3 - 25*n/12-43/4)+3^(2n)*(4*n^2-41*n/4+87/8).

G.f.: -x*(486*x^4-405*x^3-819*x^2+221*x+5) / ((x-1)*(3*x-1)^3*(9*x-1)^3). [Colin Barker, Jan 01 2013]

Maple

a := proc (n) options operator, arrow: -1/8+3^n*((2/3)*n^2-(25/12)*n-43/4)+3^(2*n)*(4*n^2-(41/4)*n+87/8) end proc: seq(a(n), n = 1 .. 18);

Mathematica

LinearRecurrence[{37, -549, 4185, -17523, 40095, -45927, 19683}, {5, 406, 11458, 221572, 3519703, 49623850, 646314724}, 20] (* Harvey P. Dale, May 21 2020 *)

Crossrefs

Cf. A221046.

Sequence in context: A198538 A198535 A128866 * A368021 A075769 A046274

Adjacent sequences: A221044 A221045 A221046 * A221048 A221049 A221050

Keyword

nonn,easy

Author

Emeric Deutsch, Dec 30 2012

Status

approved