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 A226052 Denominators of signed Egyptian fractions with sums converging to sqrt(2). 4

%I #8 Feb 23 2018 22:02:41

%S 2,11,195,180120,120479425978,27716921130006533867139,

%T 1040296455490146050257045342043017466273633682

%N Denominators of signed Egyptian fractions with sums converging to sqrt(2).

%C Using the algorithm defined at A226049 with r = sqrt(2) and f(n) = 1/n gives r = 1 + 1/2 - 1/11 + 1/195 - 1/180120 + ..., of which the 13th partial sum differs from the r by less than 10^(-2900). For a guide to related sequences, see A226049.

%H Clark Kimberling, <a href="/A226052/b226052.txt">Table of n, a(n) for n = 1..12</a>

%e Let r = sqrt(2). Then

%e 1 < r < 1 + 1/2, so a(1) = 2.

%e 1 + 1/2 -1/11 < r, so a(2) = 11.

%e 1 + 1/2 - 1/11 + 1/195 > r, so a(3) = 195.

%t \$MaxExtraPrecision = Infinity;

%t nn = 12; f[n_] := 1/n; r = Pi; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

%Y Cf. A226049.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 24 2013

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Last modified June 6 01:24 EDT 2023. Contains 363138 sequences. (Running on oeis4.)