A367795 Triangle read by rows, where row n = L(n) is defined by L(1) = [1,0] and L(n+1) is obtained from L(n) by inserting their binary concatenation between elements x,y.

1, 0, 1, 2, 0, 1, 6, 2, 4, 0, 1, 14, 6, 26, 2, 20, 4, 8, 0, 1, 30, 14, 118, 6, 218, 26, 106, 2, 84, 20, 164, 4, 72, 8, 16, 0, 1, 62, 30, 494, 14, 1910, 118, 950, 6, 1754, 218, 7002, 26, 3434, 106, 426, 2, 340, 84, 2708, 20, 5284, 164, 1316, 4, 584, 72, 1160, 8, 272, 16, 32, 0

Offset

1,4

Comments

0 is considered to be a 1-bit-long number and has 0 for binary expansion.

The numbers of bits of the numbers in this triangle are provided by the A049456 triangle.

The sorted set of the numbers that occur in some row of this triangle is provided by A367745.

Links

Table of n, a(n) for n=1..69.

Luc Rousseau, SVG illustration of the nesting of the L(n) lists for n=1..9.

Formula

Length of row n = #L(n) = 2^(n-1) + 1 = A000051(n-1).

Example

Triangle begins:
  1 0
  1 2 0
  1 6 2 4 0
  1 14 6 26 2 20 4 8 0
  1 30 14 118 6 218 26 106 2 84 20 164 ...
Or the same in binary:
  1 0
  1 10 0
  1 110 10 100 0
  1 1110 110 11010 10 10100 100 1000 0
  1 11110 1110 1110110 110 11011010 11010 1101010 10 1010100 10100 10100100 ...

Prog

(PARI)
sz(n)=if(n==0, 1, logint(n, 2)+1)
L(n)=if(n==1, List([1, 0]), my(LL=L(n-1), k=#LL); while(k>1, listinsert(LL, (LL[k-1] << sz(LL[k])) + LL[k], k); k--); LL)
for(k=1, 8, my(l=L(k)); for(i=1, #l, print1(l[i], ", ")))
(Python)
from itertools import chain, count, islice, zip_longest
def agen(): # generator of terms
    L = ["1", "0"]
    for k in count(1):
        yield from (int(t, 2) for t in L)
        Lnew = [s+t for s, t in zip(L[:-1], L[1:])]
        L = [t for t in chain(*zip_longest(L, Lnew)) if t is not None]
print(list(islice(agen(), 69))) # Michael S. Branicky, Nov 30 2023

Crossrefs

Cf. A000051, A049456, A367745.

Sequence in context: A161121 A021500 A175197 * A343825 A339031 A367270

Adjacent sequences: A367792 A367793 A367794 * A367796 A367797 A367798

Keyword

nonn,tabf,base

Author

Luc Rousseau, Nov 30 2023

Status

approved