0,1

The corresponding denominators are 4*A120785(n).

sqrt(2)+sqrt(3) = (4*sin(Pi/4) + 6*tan(Pi/6))/2 = 3.146264370 (maple10, 10 digits) This is the arithmetic mean of the areas of an 8-gon (octagon), resp. 6-gon (hexagon) inscribed, resp. circumscribed in a unit circle.

Popper (see the reference) argues that Platon new about the sum of sqrt(2)+sqrt(3). This sum approximates pi with a relative error of 1.5 permille. The two rectangular triangles, one with side lengths (1,1/2,sqrt(3)/2) and the other with side lengths (sqrt(2),1,1) are used in Platon's Timaios [53d] to build four of the five regular polyhedra (Platonic solids).

The Taylor series for sqrt(2)= sqrt(1+1) and sqrt(3)= 3*sqrt(1-2/3) are used here. Therefore lim_{n->infinity} r(n) = sqrt(2)+sqrt(3), with rationals r(n) defined below.

K. R. Popper, Die Welt des Parmenides, Piper, 2001, 2005. Ch. 8: Platon und die Geometrie (1950), pp. 326-337. English: The World of Parmenides, Routledge, London, New York, 1998.

W. Lang: Rationals r(n), limit.

a(n)= numerator(r(n)) with r(n):= 4-(sum(C(k)*(1+2^(k+1))/16^k,k=0..n)/4, n>=0, with C(k)=A000108(k) (Catalan numbers).

Rationals r(n): [13/4, 203/64, 1615/512, 51595/16384,

412529/131072, 6599099/2097152, 52788535/16777216,...].

Sequence in context: A130549 A157690 A055478 this_sequence A014404 A057807 A057804

Adjacent sequences: A121500 A121501 A121502 this_sequence A121504 A121505 A121506

nonn,easy

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