A192847 Molecular topological indices of the tetrahedral graphs.

0, 0, 0, 72, 1080, 7020, 30240, 100800, 281232, 687960, 1520640, 3100680, 5920200, 10702692, 18476640, 30663360, 49180320, 76561200, 116093952, 171978120, 249502680, 355245660, 497296800, 685504512, 931748400, 1250238600, 1657843200, 2174445000, 2823328872, 3631600980

Offset

1,4

Comments

The tetrahedral graph of order n is a vertex transitive graph with n*(n-1)*(n-2)/6 vertices and radius 3. The number of nodes at distance k=1..3 from a designated starting node are given by 3*(n-3), 3*(n-3)*(n-4)/2, (n-3)*(n-4)*(n-5)/6 respectively. - Andrew Howroyd, May 23 2017

Extended to a(1)-a(5) using the formula. - Eric W. Weisstein, Jun 26 2017

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Molecular Topological Index

Eric Weisstein's World of Mathematics, Tetrahedral Graph

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = n*(n-1)*(n-2)*(n-3)^2*(n^2-3*n+8)/4. - Andrew Howroyd, May 23 2017

From Eric W. Weisstein, Jun 26 2017: (Start)

a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8).

G.f.: 36*x^4*(2+14*x+11*x^2+8*x^3)/(1-x)^8. (End)

E.g.f.: x^4*(12 + 24*x +9*x^2 +x^3)*exp(x)/4. - G. C. Greubel, Jan 05 2018

Example

From Andrew Howroyd, May 23 2017 (Start)
Case n=8:
There are 56 vertices and the number of nodes which are at distances 1..3 from a designated starting node are 15,30,10. The molecular topological index for the graph is then 56*15*15 + 56*15*(1*15 + 2*30 + 3*10) = 100800.
(End)

Mathematica

Table[n (n - 1) (n - 2) (n - 3)^2 (n^2 - 3 n + 8)/4, {n, 20}] (* Eric W. Weisstein, Jun 26 2017 *)
Table[6 Binomial[n, 4] (n - 3) (n^2 - 3 n + 8), {n, 20}] (* Eric W. Weisstein, Jun 26 2017 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 0, 72, 1080, 7020, 30240, 100800}, 20] (* Eric W. Weisstein, Jun 26 2017 *)
CoefficientList[Series[(36 x^3 (2 + 14 x + 11 x^2 + 8 x^3))/(-1 + x)^8, {x, 0, 20}], x] (* Eric W. Weisstein, Jun 26 2017 *)

Prog

(PARI) vector(40, n, n*(n-1)*(n-2)*(n-3)^2*(n^2-3*n+8)/4) \\ G. C. Greubel, Jan 05 2019
(Magma) [n*(n-1)*(n-2)*(n-3)^2*(n^2-3*n+8)/4: n in [1..40]]; // G. C. Greubel, Jan 05 2019
(Sage) [n*(n-1)*(n-2)*(n-3)^2*(n^2-3*n+8)/4 for n in (1..40)] # G. C. Greubel, Jan 05 2019
(GAP) List([1..40], n -> n*(n-1)*(n-2)*(n-3)^2*(n^2-3*n+8)/4); # G. C. Greubel, Jan 05 2019

Crossrefs

Sequence in context: A008399 A232573 A022148 * A064567 A269088 A168194

Adjacent sequences: A192844 A192845 A192846 * A192848 A192849 A192850

Keyword

nonn

Author

Eric W. Weisstein, Jul 11 2011

Extensions

Terms a(11)-a(30) from Andrew Howroyd, May 23 2017

a(1)-a(5) added by Eric W. Weisstein, Jun 26 2017

Status

approved