A292816 b(0) = 1, b(2*n-1) = 1/(1+1^2/(1+1^2/(1+2^2/(1+2^2/(...+(n-1)^2/(1+n^2)))))) and b(2*n) = 1/(1+1^2/(1+1^2/(1+2^2/(1+2^2/(...+n^2/(1+n^2)))))). a(n) is the numerator of b(n).

1, 1, 2, 6, 14, 34, 194, 1282, 1462, 18218, 146086, 1457782, 6716878, 39074098, 45252802, 5408759762, 31474373714, 234791957218, 603801637054, 9995479925774, 3351221125294, 639914357324914, 9795281594021882, 194090616503597114, 1604611166042748122

Offset

0,3

Comments

The limit of b(n) is (PolyGamma(1,(1+sqrt(5))/4)-PolyGamma(1,(3+sqrt(5))/4))/2. See A091659.

Links

Table of n, a(n) for n=0..24.

Eric Weisstein's World of Mathematics, Polygamma Function

Eric Weisstein's World of Mathematics, Ramanujan Continued Fractions

Example

b(0) = 1/1, so a(0) = 1.
b(1) = 1/(1+1^2) = 1/2, so a(1) = 1.
b(2) = 1/(1+1^2/(1+1^2)) = 2/3, so a(2) = 2.
b(3) = 1/(1+1^2/(1+1^2/(1+2^2))) = 6/11, so a(3) = 6.
b(4) = 1/(1+1^2/(1+1^2/(1+2^2/(1+2^2)))) = 14/23, so a(4) = 14.

Crossrefs

Cf. A091659, A292817.

Sequence in context: A177790 A307068 A269506 * A105635 A178320 A297187

Adjacent sequences: A292813 A292814 A292815 * A292817 A292818 A292819

Keyword

nonn,frac

Author

Seiichi Manyama, Sep 24 2017

Status

approved