

A058227


Number of edges in all simple (loopless) paths, connecting any node with all the remaining ones in optimal graphs of degree 4.


0




OFFSET

1,1


COMMENTS

Number of edges occurring in all simple, loopless paths, connecting any node with all the remaining ones forming optimal graphs degree of 4, (2*d(G)^2+2*d(G)+1, 2*d(G)+1) with d(G) denoting graph diameter.


REFERENCES

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, AddisonWesley company, 1994.


LINKS



FORMULA

a(n) = d(V)*Sum_{k=1..d(G)} k*(2^k1) where d(V) is graph degree, and d(G) is graph diameter a(n)=d(V)*array(1..d(G))*array(1..(2^d(G)1));
a(n) = 4 * Sum_{k=1..n} k*(2^k1).
a(n) = (8*n8)*2^n  2*n^2  2*n + 8.
a(n) = a(n1) + 4*n*(2^n1) with a(0)=0.
(End)


EXAMPLE

a(5)=4(1+2*3+3*7+4*15+5*31)=972 S := array(1..5,[1,2,3,4,5]); T := array(1..5,[1,3,7,15,31]); a(5) := evalm(S&*T); a(5) := 243


MAPLE

d(V) := 4; n := 5; a(n) := d(V)*sum('n*(2^n1)', 'n'=1..n); or d(V) := 4; S := array(1..5, [1, 2, 3, 4, 5]); T := array(1..5, [1, 3, 7, 15, 31]); a(5) := d(V)*evalm(S&*T);


CROSSREFS



KEYWORD

nonn,more


AUTHOR

S. Bujnowski (slawb(AT)atr.bydgoszcz.pl), Feb 13 2002


STATUS

approved



