#### Problem

*Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:*

* *

*It can be verified that T*_{285} = P_{165} = H_{143} = 40755.

*Find the next triangle number that is also pentagonal and hexagonal.*

#### Solution

let naturalNumbers n = Seq.unfold (fun state -> Some(state, state + 1I)) n
// define the function T, P and H
let T n = n * (n + 1I) / 2I
let P n = n * (3I * n - 1I) / 2I
let H n = n * (2I * n - 1I)
// define the sequences for each function from the point the brief left off at
let TSeq = naturalNumbers 285I |> Seq.map T
let PSeq = naturalNumbers 165I |> Seq.map P
let HSeq = naturalNumbers 143I |> Seq.map H
let answer =
HSeq
|> Seq.skip 1
|> Seq.filter (fun h -> PSeq |> Seq.takeWhile (fun p -> p <= h) |> Seq.exists (fun p -> p = h))
|> Seq.filter (fun h -> TSeq |> Seq.takeWhile (fun t -> t <= h) |> Seq.exists (fun t -> t = h))
|> Seq.head