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Fungrim entry: 41f67b

2an+1=an+bn  and  bn+12=anbn   where (ak,bk)=agmk ⁣(a,b)2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)
Assumptions:nZ0  and  aC  and  bCn \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
TeX:
2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
AGMSequenceagmn ⁣(a,b)\operatorname{agm}_{n}\!\left(a, b\right) Convergents in AGM iteration
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("41f67b"),
    Formula(Where(And(Equal(Mul(2, a_(Add(n, 1))), Add(a_(n), b_(n))), Equal(Pow(b_(Add(n, 1)), 2), Mul(a_(n), b_(n)))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC