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A111510
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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).
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2
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6, 19, 90, 282, 945, 2976, 9450, 29725, 93555, 294029, 924042, 2903286, 9121612, 28657229, 90030845, 282842357, 888579011, 2791558571, 8769948430, 27551618646, 86555983553, 271923674412, 854273468992, 2683779334264
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OFFSET
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2,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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