2,1

Lim_{n->inf.} i_n/i_(n-1) approaches Pi. e.g. 2791558571/8885799011=~3.141598593...

See A108925. Analytical Pi (for n>=4 but here n>10^6 say),(n=1 2 3...n). Take n straight lines monotonically increasing in length by one and join them end to end; the last to the first. When the enclosed area is at its maximum every vertex will lie on the circumference of a circle the diameter of which divided into Triangular(n) equals Pi.

There is an interesting benchmark when n=8. The radius calculated using Pi equals 5.7296...; one tenth of the number of degrees in a radian. The radius when plotted as a drawing is very near to six and, tentatively, this could be ten times a constant near to point six.

a(n)=d where d is the integer divisor of Pi^n for even n and (Pi^n)-ne for odd n having a solution closest to Zeta(n).

a(2) = 6 then (Pi^2)/6 = Zeta(2); a(3)=19, (Pi^3-3e)/19 approx = Zeta(3); a(4)=90, (Pi^4)/90 = Zeta(4); and the only special case the author has found where ((Pi^4)-4e)/80 approx = Zeta(4).

f[n_] := Round@If[EvenQ@n, Pi^n/Zeta@n, (Pi^n - n*E)/Zeta@n]; Table[ f@n, {n, 2, 26}] (* Robert G. Wilson v *)

Cf. A108925.

Sequence in context: A123950 A026545 A041937 this_sequence A151277 A138748 A097899

Adjacent sequences: A111507 A111508 A111509 this_sequence A111511 A111512 A111513

nonn

Marco Matosic (marcomatosic(AT)hotmail.com), Nov 16 2005

Corrected and extended by Robert G. Wilson v (rgwv(at)rgwv.com), Nov 18 2005

Corrections from Marco Matosic (marcomatosic(AT)hotmail.com), Mar 27 2006

Definition clarified by Omar E. Pol (info(AT)polprimos.com), Jan 02 2009