A334955 Number of 1's in n-th string when A000002 is expressed as 1, 2, 2, 11, 21, 221, 22112, 11221211, 21221121121, 2212211212212112, ....

0, 2, 1, 1, 2, 5, 6, 7, 12, 19, 29, 43, 62, 95, 148, 216, 322, 482, 727, 1083, 1627, 2457, 3660, 5518, 8250, 12403, 18594, 27914, 41863, 62781, 94158, 141205, 211797, 317782, 476743, 714948, 1072526, 1608732, 2413223, 3619779, 5428986, 8144054

Offset

3,2

Comments

Representing A000002 as a tree. Each branch of this tree is a string. Starting from n=3, each 1 in n-th string generates either 1 or 2 in (n+1)-th string and each 2 in n-th string generates either 11 or 22 in (n+1)-st string based on the previously generated term of either 2 or 1. Hence number of terms in (n+1)-st string is sum of all terms in n-th string.

Links

Table of n, a(n) for n=3..44.

Prog

(Python)
MAX_NUM = 10000 # Number of terms in Kolakoski Sequence
K = [1, 2, 2]
previous = 2
# Generate Kolakoski Sequence
for index1 in range(2, MAX_NUM):
    generator = K[index1]
    if (generator == 1 and previous == 1):
        K.append(2)
        previous = 2
    elif(generator == 2 and previous == 1):
        K.append(2)
        K.append(2)
        previous = 2
    elif(generator == 1 and previous == 2):
        K.append(1)
        previous = 1
    elif(generator == 2 and previous == 2):
        K.append(1)
        K.append(1)
        previous = 1
branch_sum = 1
index2 = 2
cntr1 = 1
while(index2 < MAX_NUM):
    ones_cntr = 0
    # This for loop extracts strings from Kolakoski Sequence
    start_index = index2
    end_index = index2 + branch_sum
    branch_sum = 0
    for index3 in range(start_index , end_index):
        branch_sum = branch_sum + K[index3]
        ones_cntr = ones_cntr + (K[index3]%2)
    index2 = end_index
    print(str(cntr1)+"   "+str(ones_cntr)+"\n")
    cntr1 = cntr1 + 1

Crossrefs

Cf. A000002.

First differences of A111124.

Sequence in context: A172483 A216396 A273488 * A117848 A025242 A356696

Adjacent sequences: A334952 A334953 A334954 * A334956 A334957 A334958

Keyword

nonn

Author

Rakesh Khanna A, May 24 2020

Extensions

Missing a(42) inserted and a(44) added by Andrei Zabolotskii, Oct 09 2025

Status

approved