A079668 Start with 1; at n-th step, write down what is in the sequence so far.

1, 1, 1, 3, 1, 4, 1, 0, 2, 1, 3, 1, 0, 6, 1, 1, 2, 2, 3, 1, 4, 2, 0, 10, 1, 3, 2, 3, 3, 2, 4, 0, 5, 1, 6, 4, 0, 12, 1, 6, 2, 6, 3, 3, 4, 1, 5, 2, 6, 0, 7, 0, 8, 0, 9, 1, 10, 8, 0, 15, 1, 8, 2, 8, 3, 5, 4, 2, 5, 5, 6, 1, 7, 1, 8, 1, 9, 2, 10, 0, 11, 1, 12, 10

Offset: 1

Author

N. J. A. Sloane Jan 26 2003

Keywords

• nonn
• base
• tabf
• changed

Comments

After 1 1 1 3 1, we see "4 1's, 0 2's and 1 3", so next terms are 4 1 0 2 3 1.

Examples

			Row n lists all terms written at the n-th step:
  1;
  1, 1;
  3, 1;
  4, 1,  0, 2, 1, 3;
  1, 0,  6, 1, 1, 2, 2, 3, 1, 4;
  2, 0, 10, 1, 3, 2, 3, 3, 2, 4, 0, 5, 1, 6;
  4, 0, 12, 1, 6, 2, 6, 3, 3, 4, 1, 5, 2, 6, 0, 7, 0, 8, 0, 9, 1, 10;
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20377 (first 55 steps)

Crossrefs

For other versions see A051120, A055186, A079686.

Cf. A030717, A333867.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+add(x^i, i=T(n))) end:
T:= proc(n) option remember; `if`(n=1, 1, (p->
      seq([coeff(p, x, i), i][], i=ldegree(p)..degree(p)))(b(n-1)))
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 24 2025

Mathematica

s={1}; Do[s=Flatten[{s,{Count[s,#],#}&/@Range[Min[s],Max[s]]}],{20}];s (* Peter J. C. Moses, Mar 21 2013 *)

Python

from itertools import islice
def agen(): # generator of terms
    a = [1]
    yield 1
    while True:
        counts = []
        for d in range(min(a), max(a)+1):
            c = a.count(d)
            counts.extend([c, d])
        a += counts
        yield from counts
print(list(islice(agen(), 84))) # Michael S. Branicky, Aug 24 2025

Extensions

More terms from Sean A. Irvine, Aug 24 2025