1. Generalizations:
(G) Take c(0)=b and c(n)='frequency' of digits in k-th power of sum of digits of c(n-1) followed by 'digit'-indication.
(1.G) c(n)=A045918(j)=A045918(i+A118881(m)), where i=0,1,...,k-1 and k is already chosen in (G), i depends on A118881(m) and c(n) can be represented using ordered pair (n,j,i,m).
Example: a(n)=A045918(j)=A045918(i+A118881(m)), where i=0 or 1 depends on A118881(m) and with ordered pair (n,j,i,m) explains the a(n) term as follows:
a(0)=1,  (1,-,-,1), i and j does not exist for n=0,
a(1)=11,     (2,1,0,1)
a(2)=14,     (3,4,2,0)
a(3)=1215,   (3,26,1,5)
a(n)=1811,   (n,82,1,9)  for n=4+6*s, s=0,1,...
a(n)=111211, (n,122,1,9) for n=5+6*s, s=0,1,...
a(n)=1419,   (n,50,1,9)  for n=6+6*s, s=0,1,...
a(n)=2215,   (n,226,1,9) for n=7+6*s, s=0,1,...
a(n)=1120,   (n,121,1,9) for n=8+6*s, s=0,1,...
a(n)=1116,   (n,37,1,9)  for n=9+6*s, s=0,1,..
The above sequence is for a(0)=1 and k=2.

2. Variation:
(V1) Take c(0)=b and c(n)=summarize the 'frequency' of digits in k-th power of sum of digits of c(n-1) followed by 'digit'-indication in increasing order, in the manner of A005151, e.g., c(0)=1, k=2 then c(0)=1 has 1 digit, so sum of digits is 1 and square of sum of digits is 1. So, c(1) = 11 that is one times 1. As c(1)=11 has 2 digits, so the sum of digits is 1+1=2 and the square of the sum of digits is 4. So, c(2) = 14, that is, one times 4, and so on; c(5)=1811 has 4 digits, so the sum of digits is 1+8+1+1=11 and the square of the sum of digits is 121. So, c(6) = 2112 that is two times one and then one times two occurred in expansion of c(6).
(V2) Take c(0)=b and c(n)=summarize the 'frequency' of digits in k-th power of sum of digits of c(n-1) followed by 'digit'-indication in decreasing order, in the manner of A007890, e.g., c(0)=1, k=2 then c(0)=1 has 1 digit, so the sum of digits is 1 and the square of the sum of digits is 1. So, c(1) = 11 that is one times 1. As c(1)=11 has 2 digits, so the sum of digits is 1+1=2 and the square of the sum of digits is 4. So, c(2) = 14 that is one times 4 and c(2)=14 has 2 digits, so the sum of digits is 1+4=5 and the square of the sum of digits is 25. So, c(6) = 1512, that is, two times one and then one times two occurred in expansion of c(6).

3. Formula, G, V1 and V2 holds when k and b are natural numbers.