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
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