https://www.wolframalpha.com/input?i=lim%28x-%3Einf%29+%28%281%2B%CF%80%5E4*2%5E%28-4x-2%29+-%CF%80%5E2%2F2%5E%282x%29%29+%2F+%28cos%282%CF%80%2F2%5Ex%29+%2B1%29+%2B%E2%88%9A5%2F2%29&lang=ja

Ramanujan's Notebooks IV p353
https://archive.org/details/ramanujans-notebooks/Ramanujan%27s%20Notebooks%20IV/page/n181/mode/2up
に載っている黄金比を使うπの公式の高次版

1/π = 10(√√5)Σ[n=0,∞] (((2n)!)^3/(64^n(n!)^6))(1+6n((1+√5)/2)^2)((1+√5)/2)^(-24n-8)

ラマヌジャンはなぜこの式を載せなかったのだろうか？

黄金比の級数表示
https://www.wolframalpha.com/input?i=sum+%282n%2B3%29%21%2F%28n%21%28n%2B2%29%215%5E%28n%2B2%29%29+from+n%3D0+to+infinity&lang=ja

和の前の係数はフィボナッチ数の比

>黄金比の級数表示
これも追加

あまり知られてない高次の黄金比を含むRogers-Ramanujan連分数

二式目左辺
https://www.wolframalpha.com/input?i=e%5E%28-2pi%29+ContinuedFractionK%5Be%5E%28-10+n+pi%29%2C+1%2C+%7Bn%2C+0%2C+20%7D%5D&lang=ja
二式目右辺
https://www.wolframalpha.com/input?i=%281%2Bgoldenratio%5E2%29%2F%28goldenratio%2B%283+sqrt%281%2Bgoldenratio%5E2%29-4-goldenratio%29%5E%281%2F5%29%29-goldenratio&lang=ja