2 citations · 6 across the 10 of their papers we have counts for
10 papers
On the formation of continued fractions
Leonhard Euler
In this paper Euler shows how, if we have recursive functions f,g,h and an infinite sequence A,B,C,... which satisfies fA=gB+hC, f'B=g'C+h'D, f''C=g''D+h''E, f'''D=g'''E+h'''F, etc…
On a new type of rational and highly convergent series, by which the ratio of the circumference to the diameter is able to be expressed
Leonhard Euler
In this short paper Euler gives a highly convergent series for arctan and thus pi, which converges much faster than the Leibniz series for arctan.
A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers
Leonhard Euler
Euler gives a continued fraction representation of (1+x)^n involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fra…
An illustration of a paradox about the idoneal, or suitable, numbers
Leonhard Euler
In this paper Euler shows that there are no additional square idoneal numbers aside from 1, 4, 9, 16, and 25.
On the expansion of the power of any polynomial 1+x+x^2+x^3+x^4+etc
Leonhard Euler
In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coe…
On the remarkable properties of the pentagonal numbers
Leonhard Euler
In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form . He recalls that the infinite product $(1-x)(1-x^2)(1-x^3).…