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

0,2

Comments

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

Links

Table of n, a(n) for n=0..20.

Formula

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

Example

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.

Prog

(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
}

Crossrefs

Cf. A000002. Counterpart of A195206.

Sequence in context: A369295 A145641 A317782 * A195211 A106415 A212819

Adjacent sequences: A342404 A342405 A342406 * A342408 A342409 A342410

Keyword

nonn

Author

Ishan Goel, Mar 11 2021

Status

approved