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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). 2
6, 19, 90, 282, 945, 2976, 9450, 29725, 93555, 294029, 924042, 2903286, 9121612, 28657229, 90030845, 282842357, 888579011, 2791558571, 8769948430, 27551618646, 86555983553, 271923674412, 854273468992, 2683779334264 (list; graph; refs; listen; history; text; internal format)



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


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


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, Nov 18 2005 *)


Cf. A100594, A108925.

Sequence in context: A026545 A041937 A279512 * A151277 A192368 A285853

Adjacent sequences:  A111507 A111508 A111509 * A111511 A111512 A111513




Marco Matosic, Nov 16 2005


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



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Last modified July 26 06:03 EDT 2017. Contains 289798 sequences.