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

%I A334955 #47 Jun 01 2025 17:40:22
%S A334955 0,2,1,1,2,5,6,7,12,19,29,43,62,95,148,216,322,482,727,1083,1627,2457,
%T A334955 3660,5518,8250,12403,18594,27914,41863,62781,94158,141205,211797,
%U A334955 317782,476743,714948,1072526,1608732,2413223,5428986
%N A334955 Number of 1's in n-th string when A000002 is expressed as 1, 2, 2, 11, 21, 221, 22112, 11221211, 21221121121, 2212211212212112, ....
%C A334955 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.
%o A334955 (Python)
%o A334955 MAX_NUM = 10000 # Number of terms in Kolakoski Sequence
%o A334955 K = [1,2,2]
%o A334955 previous = 2
%o A334955 # Generate Kolakoski Sequence
%o A334955 for index1 in range(2,MAX_NUM):
%o A334955     generator = K[index1]
%o A334955     if (generator == 1 and previous == 1):
%o A334955         K.append(2)
%o A334955         previous = 2
%o A334955     elif(generator == 2 and previous == 1):
%o A334955         K.append(2)
%o A334955         K.append(2)
%o A334955         previous = 2
%o A334955     elif(generator == 1 and previous == 2):
%o A334955         K.append(1)
%o A334955         previous = 1
%o A334955     elif(generator == 2 and previous == 2):
%o A334955         K.append(1)
%o A334955         K.append(1)
%o A334955         previous = 1
%o A334955 branch_sum = 1
%o A334955 index2 = 2
%o A334955 cntr1 = 1
%o A334955 while(index2 < MAX_NUM):
%o A334955     ones_cntr = 0
%o A334955     # This for loop extracts strings from Kolakoski Sequence
%o A334955     start_index = index2
%o A334955     end_index = index2 + branch_sum
%o A334955     branch_sum = 0
%o A334955     for index3 in range(start_index , end_index):
%o A334955         branch_sum = branch_sum + K[index3]
%o A334955         ones_cntr = ones_cntr + (K[index3]%2)
%o A334955     index2 = end_index
%o A334955     print(str(cntr1)+"   "+str(ones_cntr)+"\n")
%o A334955     cntr1 = cntr1 + 1
%Y A334955 Cf. A000002.
%K A334955 nonn
%O A334955 3,2
%A A334955 _Rakesh Khanna A_, May 24 2020