A342407 Number of 2's in the first 10^n entries of the Kolakoski sequence (A000002).

0, 5, 51, 498, 5004, 50028, 500014, 4999954, 49999325, 499998777, 5000002329, 49999998413, 499999949299, 4999999991841, 49999999683763, 499999999022579, 5000000005362272, 50000000022520652, 500000000055534895, 5000000000274296550, 50000000000909149240

Offset: 0

Author

Ishan Goel, Mar 11 2021

Keywords

• nonn

Comments

This is a counterpart of A195206 in the sense that A195206 lists the number of 1's.

Examples

			The first 10 entries of the Kolakoski sequence (A000002) are 1221121221. From this we see that a(0) = 0, since the first term is not a 2, and a(1)=5 since among the first 10 terms, 5 of them are 2's.
		

Crossrefs

Cf. A000002. Counterpart of A195206.

Programs

Go

package main
import (
    "fmt"
    "math"
)
func main() {
    fmt.Println(a(5))
}
func a(n int) int {
    k := int(math.Pow(10, float64(n))) // get the number of terms of the Kolakoski sequence (A000002) to generate
    // seq stores the Kolakoski sequence
    seq := make([]int, 0, k+1) // n+1 because instruction 2 could add one extra
    var ind, i, toAppend int   // ind represents which instruction to follow
    seq = append(seq, 1, 2, 2) // initial terms
    ind = 2                    // follow third instruction next
    i += 3                     // three numbers already added
    for i < k {
        if seq[ind] == 1 { // add one number?
            if seq[len(seq)-1] == 1 {
                toAppend = 2
            } else {
                toAppend = 1
            }
            seq = append(seq, toAppend)
            i++   // we added a number
            ind++ // next instruction
        } else { // add two numbers
            if seq[len(seq)-1] == 1 {
                toAppend = 2
            } else {
                toAppend = 1
            }
            seq = append(seq, toAppend, toAppend) // append two numbers
            i += 2 // we added two numbers
            ind++
        }
    }
    seq = seq[:k] // trim to k values
    // now count twos
    var twos int
    for _, curr := range seq {
        if curr == 2 {
            twos++
        }
    }
    return twos
}

Formula

a(n) = 10^n - A195206(n).