A221043 - OEIS

%I #17 Feb 13 2024 10:21:02

%S 10,457,11788,223306,3527782,49658659,646456696,7958918644,

%T 94000489378,1075247030365,11991524116804,131012134626814,

%U 1407240945512638,14901372361780855,155885329216404592,1613748977026119016,16554187553043529402,168462466522953130609

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

%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) = -(7/8)+3^n*(2*n^2-(9/4)*n-10)+3^(2*n)*(4*n^2-(41/4)*n+(87/8)).

%F G.f.: x*(243*x^4+3807*x^3-369*x^2-87*x-10) / ((x-1)*(3*x-1)^3*(9*x-1)^3). [_Colin Barker_, Jan 01 2013]

%p a := proc (n) options operator, arrow: -7/8+3^n*(2*n^2-(9/4)*n-10)+3^(2*n)*(4*n^2-(41/4)*n+87/8) end proc: seq(a(n), n = 1 .. 18);

%Y Cf. A221042.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Dec 30 2012

%E Offset changed from 0 to 1 by _Bruno Berselli_, Dec 30 2012