A076744 Related to the expected length of the shortest nonintersecting path through n points on a Sierpiński Gasket from corner to corner.

4, 72, 4176, 731808, 381879360, 592267282560, 2733202405059840, 37590062966534453760, 1542797317119230338360320, 189160927199005707074274969600, 69339320764681731806436884240486400, 76034016779649931607383549266935620608000

Offset: 1

Author

Nathan B. Shank, Nov 11 2002

Keywords

• nonn

Comments

Original name: This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest nonintersecting path through n points on a Sierpiński Gasket from corner to corner.

Does b(n) / n^(1-log(2)/log(3)) -> constant?

Programs

Maple

b:=n->sum(binomial(n,k)*(-1)^k/(2*3^k-3),k=0..n);
c:=n->product(2*3^k-3,k=0..n);
a:=n->b(n)*c(n);
seq(a(n),n=1..12);

Formula

a(n) = b(n) * c(n) where b(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) / (2*3^k-3) and c(n) = Product_{k=0..n} (2*3^k-3).

Extensions

Revised by Sean A. Irvine, Apr 14 2025