A227709 -id:A227709

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A227707

The terminal Wiener index of the dendrimer D_n defined pictorially in Fig. 1 of the Heydari et al. reference.

+10
3

12, 78, 444, 2328, 11568, 55392, 258240, 1180032, 5309184, 23594496, 103812096, 452990976, 1962946560, 8455741440, 36238835712, 154618920960, 657130192896, 2783139201024, 11751031308288, 49478024822784, 207807700795392, 870813215490048, 3641582523777024, 15199648767541248

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

REFERENCES

A. Heydari, I. Gutman, On the terminal index of thorn graphs, Kragujevac J. Sci., 32, 2010, 57-64.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (10, -32, 32).

FORMULA

a(n) = 3*2^n + 9*4^n*(n+1).

G.f.: 6*(2-7*x+8*x^2)/((1-2*x)*(1-4*x)^2).

MAPLE

a := proc (n) options operator, arrow: 3*2^n+9*4^n*(n+1) end proc: seq(a(n), n = 0 .. 25);

MATHEMATICA

CoefficientList[Series[6 (2 - 7 x + 8 x^2) / ((1 - 2 x) (1 - 4 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 04 2013 *)

PROG

(Magma) [3*2^n+9*4^n*(n+1): n in [0..25]]; // Vincenzo Librandi, Aug 04 2013

CROSSREFS

Cf. A227708, A227709.

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Aug 02 2013

STATUS

approved

A227708

The Wiener index of the dendrimer D_n defined pictorially in Fig. 1 of the Heydari et al. reference.

+10
3

84, 354, 1674, 8178, 39858, 191250, 900498, 4164114, 18952722, 85106706, 377862162, 1661755410, 7249502226, 31410683922, 135299432466, 579837493266, 2473936945170, 10514155438098, 44530379784210, 188016821796882, 791649070350354

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

a(2) has been checked by the direct computation of the distance matrix (with Maple).

REFERENCES

A. Heydari, I. Gutman, On the terminal index of thorn graphs, Kragujevac J. Sci., 32, 2010, 57-64.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (13,-64,148,-160,64).

FORMULA

a(n) = 18 + 2^n*(66+30*n) + 36*n*4^n.

G.f.: 6*(14-123*x+408*x^2-560*x^3+288*x^4)/((1-x)*(1-2*x)^2*(1-4*x)^2).

a(0)=84, a(1)=354, a(2)=1674, a(3)=8178, a(4)=39858, a(n)=13*a(n-1)- 64*a(n-2)+ 148*a(n-3)- 160*a(n-4)+64*a(n-5). - Harvey P. Dale, Jan 09 2016

MAPLE

a := proc (n) options operator, arrow: 18+2^n*(66+30*n)+36*4^n*n end proc: seq(a(n), n = 0 .. 25);

MATHEMATICA

CoefficientList[Series[6 (14 - 123 x + 408 x^2 - 560 x^3 + 288 x^4) / ((1 - x) (1 - 2 x)^2 (1 - 4 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 04 2013 *)

LinearRecurrence[{13, -64, 148, -160, 64}, {84, 354, 1674, 8178, 39858}, 30] (* Harvey P. Dale, Jan 09 2016 *)

PROG

(Magma) [18+2^n*(66+30*n)+36*n*4^n: n in [0..25]]; // Vincenzo Librandi, Aug 04 2013

CROSSREFS

Cf. A227707, A227709.