A058227 - OEIS

%I #14 Jul 30 2022 08:18:03

%S 4,28,112,352,972,2484,6040,14200

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

%C 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.

%D R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley company, 1994.

%F a(n) = d(V)*Sum_{k=1..d(G)} k*(2^k-1) 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));

%F Conjectures from _Sean A. Irvine_, Jul 29 2022: (Start)

%F a(n) = 4 * Sum_{k=1..n} k*(2^k-1).

%F a(n) = (8*n-8)*2^n - 2*n^2 - 2*n + 8.

%F a(n) = a(n-1) + 4*n*(2^n-1) with a(0)=0.

%F (End)

%e 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

%p d(V) := 4; n := 5; a(n) := d(V)*sum('n*(2^n-1)','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);

%K nonn,more

%O 1,1

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