A121503 Numerators of partial sums of a series for sqrt(2) + sqrt(3) involving Catalan numbers.

13, 203, 1615, 51595, 412529, 6599099, 52788535, 3378355987, 27026481101, 432421205841, 3459361042977, 110699432952143, 885595037556565, 14169517557800915, 113356129507566775, 14509583941597490435

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• easy
• frac

Comments

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 Plato knew about the sum of sqrt(2)+sqrt(3). This sum approximates Pi with a relative error of 0.15%. The two right triangles, one with side lengths (1,1/2,sqrt(3)/2) and the other with side lengths (sqrt(2),1,1) are used in Plato'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->oo} r(n) = sqrt(2)+sqrt(3), with rationals r(n) defined below.

Examples

			Rationals r(n): [13/4, 203/64, 1615/512, 51595/16384, 412529/131072, 6599099/2097152, 52788535/16777216,...].
		

References

• 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.

Links

• Wolfdieter Lang, Rationals r(n), limit.

Crossrefs

Cf. A000108, A120785.

Programs

PARI

a(n) = numerator(4 - sum(k=0, n, binomial(2*k,k)/(k+1)*(1+2^(k+1))/16^k)/4); \\ Michel Marcus, Sep 20 2023

Formula

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