3,1

Analog of A121500 with n and m roles interchanged.

For a regular m-gon circumscribed around a unit circle (area pi) the arithmetic mean of the areas of this m-gon with a regular inscribed n-gon is nearest to pi for n=a(m).

This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with a 1.5 permille relative error. See the Popper reference under A121503.

a(m)=min(abs(F(n,m)),n=3..infinity), m>=3 (checked for n=3..3+500), with F(nm):= ((Fin(n)+Fout(m))/2-Pi)/Pi), where Fin(n):=(n/2)*sin(2*Pi/n) and Fout(m):= m*tan(Pi/m). Fin(n) is the area of the regular n-gon inscribed in the unit circle. Fout(n) is the area of an regular n-gon circumscribing the unit circle. E(n,m) = (F(n,m)-pi)/pi is the relative error.

m=15, a(15)=21=n: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error E(21,15)= 0.0000147(rounded).

m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).

This is larger than E(8,6), therefore the m value 7 does not appear in A121502.

m=6, a(6)=8=n: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error E(8,6)= 0.001487 (rounded). All other inscribed n-gons with circumscribed 6-gon lead to a larger relative error.

Cf. A121502 (values for m for which relative errors E(n, m) decrease).

Sequence in context: A133561 A095117 A089585 this_sequence A114119 A101358 A047446

Adjacent sequences: A121503 A121504 A121505 this_sequence A121507 A121508 A121509

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 2006