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A121503
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Numerators of partial sums of a series for sqrt(2) + sqrt(3) involving Catalan numbers.
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6
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13, 203, 1615, 51595, 412529, 6599099, 52788535, 3378355987, 27026481101, 432421205841, 3459361042977, 110699432952143, 885595037556565, 14169517557800915, 113356129507566775, 14509583941597490435
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OFFSET
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0,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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Rationals r(n): [13/4, 203/64, 1615/512, 51595/16384, 412529/131072, 6599099/2097152, 52788535/16777216,...].
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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STATUS
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approved
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