cp's OEIS Frontend

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).

%I A111510 #16 May 29 2017 03:17:34
%S A111510 6,19,90,282,945,2976,9450,29725,93555,294029,924042,2903286,9121612,
%T A111510 28657229,90030845,282842357,888579011,2791558571,8769948430,
%U A111510 27551618646,86555983553,271923674412,854273468992,2683779334264
%N 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).
%C A111510 Lim_{n->inf.} i_n/i_(n-1) approaches Pi. e.g. 2791558571/8885799011=~3.141598593...
%C A111510 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.
%C A111510 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.
%C A111510 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
%H A111510 Vincenzo Librandi, <a href="/A111510/b111510.txt">Table of n, a(n) for n = 2..1000</a>
%e A111510 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).
%e A111510 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).
%t A111510 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 *)
%Y A111510 Cf. A100594, A108925.
%K A111510 nonn
%O A111510 2,1
%A A111510 _Marco Matosic_, Nov 16 2005
%E A111510 Corrected and extended by _Robert G. Wilson v_, Nov 18 2005
%E A111510 Corrections from _Marco Matosic_, Mar 27 2006
%E A111510 Definition clarified by _Omar E. Pol_, Jan 02 2009