A270580 - OEIS

%I #9 Feb 24 2018 14:03:32

%S 1,2,7,43,2233,5100361,40162526999265,25631935256046376027999327548,

%T 973579151885397220180400699680033378225854987721289580493,

%U 20355636044566797478491707686529410726939762602606154042023303177125252037523393842033572704449460687246942494130101

%N Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(k+1).

%C Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1).  Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k).  Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

%C See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270580/b270580.txt">Table of n, a(n) for n = 1..12</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e sqrt(1/2) = 1/(2*1) + 1/(3*2) + 1/(4*7) + 1/(5*43) + ...

%t r[k_] := 1/(k+1); f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

%Y Cf. A269993.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Apr 03 2016