A221047 - OEIS

%I #21 Feb 15 2024 11:59:55

%S 5,406,11458,221572,3519703,49623850,646314724,7958362600,93998378377,

%T 1075239211294,11991495728998,131012033254444,1407240588517147,

%U 14901371119404658,155885324936843080,1613748962415344464,16554187503550561933,168462466356459175462

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

%H Ray Chandler, <a href="/A221047/b221047.txt">Table of n, a(n) for n = 1..1000</a>

%H K. Balasubramanian, <a href="https://doi.org/10.1007/BF01170006">Recent developments in tree-pruning methods and polynomials for cactus graphs and trees</a>, J. Math. Chemistry, 4 (1990) 89-102.

%H H. Hosoya and K. Balasubramanian, <a href="https://doi.org/10.1007/BF00529932">Exact dimer statistics and characteristic polynomials of cacti lattices</a>, Theor. Chim. Acta 76 (1989) 315-329. Also on <a href="https://www.researchgate.net/publication/225247673_Exact_dimer_statistics_and_characteristic_polynomials_of_cacti_lattices">ResearchGate</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (37, -549, 4185, -17523, 40095, -45927, 19683).

%F 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).

%F 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]

%p 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);

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

%Y Cf. A221046.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Dec 30 2012