A308092 The sum of the first n terms of the sequence is the concatenation of the first n bits of the sequence read as binary, with a(1) = 1.

1, 2, 3, 7, 14, 28, 56, 112, 224, 448, 896, 1791, 3583, 7166, 14332, 28663, 57326, 114653, 229306, 458612, 917223, 1834446, 3668892, 7337785, 14675570, 29351140, 58702279, 117404558, 234809116, 469618232, 939236465, 1878472930, 3756945860, 7513891719

Offset: 1

Author

Peter Kagey, May 12 2019

Keywords

• nonn
• base

Comments

In binary, the sequence begins 1, 10, 11, 111, 1110, 11100, 111000, 1110000, 11100000, 111000000, 1110000000, 11011111111, 110111111111, 1101111111110, 11011111111100, ...

Conjecture: The number of 1's in the binary representation of each term is weakly increasing, i.e., A000120(a(n)) >= A000120(a(n-1)).

Proved by Matthew Scroggs; see link. - Peter Kagey, Jun 19 2019

Examples

			For n=5, 1 + 2 + 3 + 7 + 14 = 1_2 + 10_2 + 11_2 + 111_2 + 1110_2 = 11011_2, the first five bits of the sequence.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..1000
• Matthew Scroggs, Number of 1s in A308092

Crossrefs

Cf. A000120, A300000 (decimal analog).

Programs

Mathematica

a[1]=1;a[2]=2;a[n_]:=a[n]=FromDigits[Flatten[IntegerDigits[#,2]&/@Table[a[k],{k,n-1}]][[;;n]],2]-Total@Table[a[m],{m,n-1}]
Table[a[l],{l,40}] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

Python

def aupton(terms):
  alst, bstr = [1, 2], "110"
  for n in range(3, terms+1):
    an = int(bstr[:n], 2) - int(bstr[:n-1], 2)
    alst, bstr = alst + [an], bstr + bin(an)[2:]
  return alst
print(aupton(34)) # Michael S. Branicky, Mar 30 2021

Ruby

def first_bits(n, seq); seq.map { |i| i.to_s(2) }.join[0...n].to_i(2) end
def next_term(n, seq); first_bits(n,seq) - first_bits(n-1,seq) end
def a308092_list(n)
  (3..n).reduce([1,2]) { |accum, i| accum << next_term(i, accum) }
end

Formula

a(n) = c(n) - c(n-1) for n > 2, where c(n) is the concatenation of the first n bits of the sequence.