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A121503 Numerators of partial sums of a series for sqrt(2) + sqrt(3) involving Catalan numbers. 6
13, 203, 1615, 51595, 412529, 6599099, 52788535, 3378355987, 27026481101, 432421205841, 3459361042977, 110699432952143, 885595037556565, 14169517557800915, 113356129507566775, 14509583941597490435 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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->infinity} r(n) = sqrt(2)+sqrt(3), with rationals r(n) defined below.

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

Table of n, a(n) for n=0..15.

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

FORMULA

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).

EXAMPLE

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

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

CROSSREFS

Sequence in context: A297634 A305147 A055478 * A014404 A057807 A057804

Adjacent sequences:  A121500 A121501 A121502 * A121504 A121505 A121506

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Aug 16 2006

STATUS

approved

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Last modified April 25 12:04 EDT 2019. Contains 322456 sequences. (Running on oeis4.)