A111510 If n is even then a(n) is the nearest integer to Pi^n/Zeta(n), otherwise a(n) is the nearest integer to (Pi^n - n*e)/Zeta(n).

6, 19, 90, 282, 945, 2976, 9450, 29725, 93555, 294029, 924042, 2903286, 9121612, 28657229, 90030845, 282842357, 888579011, 2791558571, 8769948430, 27551618646, 86555983553, 271923674412, 854273468992, 2683779334264

Offset: 2

Author

Marco Matosic, Nov 16 2005

Keywords

• nonn

Comments

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.

It appears that a(2n-1) taken when rounded down (rather than to the nearest integer) is equal to A100594(n). - Terry D. Grant, May 28 2017

Examples

			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).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A100594, A108925.

Programs

Mathematica

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, Nov 18 2005 *)

Extensions

Corrected and extended by Robert G. Wilson v, Nov 18 2005

Corrections from Marco Matosic, Mar 27 2006

Definition clarified by Omar E. Pol, Jan 02 2009