加法展開を繰り返して得た諸公式から、係数の比較的大きなものをピックアップしてみた。
 
 

３次式

π/4 = 2arctan(1/2)-arctan(1/7)
π/4 = 2arctan(1/3)+arctan(1/7)
 

４次式

π/4 = 2arctan(1/2)-arctan(1/6)+arctan(1/43)
π/4 = 2arctan(1/2)-arctan(1/8)-arctan(1/57)
π/4 = 2arctan(1/2)-arctan(1/9)-arctan(1/32)
π/4 = 2arctan(1/2)-arctan(1/12)-arctan(1/17)
π/4 = 2arctan(1/3)+arctan(1/5)-arctan(1/18)
π/4 = 2arctan(1/3)+arctan(1/6)-arctan(1/43)
π/4 = 2arctan(1/3)+arctan(1/8)+arctan(1/57)
π/4 = 2arctan(1/3)+arctan(1/9)+arctan(1/32)
π/4 = 2arctan(1/3)+arctan(1/12)+arctan(1/17)

５次式

π/4 = 4arctan(1/5)-arctan(1/239)

π/4 = 3arctan(1/3)+arctan(1/57)-arctan(1/5)
π/4 = 3arctan(1/4)+arctan(1/13)-arctan(1/38)
π/4 = 3arctan(1/4)+arctan(1/20)+arctan(1/1985)

π/4 = 2arctan(1/3)+2arctan(1/12)-arctan(1/41)
π/4 = 2arctan(1/3)+2arctan(1/14)-arctan(1/1393)
π/4 = 2arctan(1/3)+2arctan(1/17)+arctan(1/41)
π/4 = 2arctan(1/4)+2arctan(1/13)+arctan(1/7)
π/4 = 2arctan(1/5)+2arctan(1/8)+arctan(1/7)

６次式

π/4 = 4arctan(1/5)+arctan(1/99)-arctan(1/70)
π/4 = 4arctan(1/5)+arctan(1/1958)-arctan(1/213)
π/4 = 4arctan(1/5)+arctan(1/4155)-arctan(1/226)
 

π/4 = 3arctan(1/3)+2arctan(1/7)-arctan(1/2)
π/4 = 3arctan(1/5)+2arctan(1/8)-arctan(1/18)
π/4 = 3arctan(1/8)+2arctan(1/5)+arctan(1/57)

７次式

π/4 = 4arctan(1/3)+2arctan(1/7)-arctan(1/1)
π/4 = 4arctan(1/5)+2arctan(1/99)-arctan(1/41)
π/4 = 4arctan(1/8)+2arctan(1/7)+arctan(1/239)

π/4 = 3arctan(1/5)+2arctan(1/21)+arctan(1/13)+arctan(1/47)
π/4 = 3arctan(1/5)+2arctan(1/47)+arctan(1/8)+arctan(1/38)
π/4 = 3arctan(1/8)+2arctan(1/57)+arctan(1/3)+arctan(1/18)
π/4 = 3arctan(1/13)+2arctan(1/4)+arctan(1/21)+arctan(1/57)
π/4 = 3arctan(1/13)+2arctan(1/7)+arctan(1/4)+arctan(1/38)

π/4 = 3arctan(1/7)+2arctan(1/6)+2arctan(1/68)
π/4 = 3arctan(1/7)+2arctan(1/8)+2arctan(1/18)

π/4 = 2arctan(1/3)+2arctan(1/17)+2arctan(1/99)+arctan(1/239)
π/4 = 2arctan(1/3)+2arctan(1/21)+2arctan(1/47)+arctan(1/239)
π/4 = 2arctan(1/3)+2arctan(1/22)+2arctan(1/75)+arctan(1/41)
π/4 = 2arctan(1/3)+2arctan(1/27)+2arctan(1/46)+arctan(1/41)
π/4 = 2arctan(1/4)+2arctan(1/8)+2arctan(1/47)+arctan(1/239)

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