A304274 The concatenation of the first n elements is the largest positive even number with n digits when written in base 3/2.

2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Author

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

Keywords

• nonn
• base

Comments

This sequence is possible due to the fact that the largest even integers are prefixes of each other.

A304272(n) is the largest even integer with n digits.

Examples

			Number 8 in base 3/2 is 212, and it is the largest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the largest even integer with 2 digits in base 3/2.
		

Links

• B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Crossrefs

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304273, A305497, A305498.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, 2,
      (t-> t+irem(t, 2))(b(n-1)*3/2))
    end:
a:= n-> b(n+1)-3/2*b(n)+1:
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 21 2018

Mathematica

b[n_] := b[n] = If[n == 1, 2, Function[t, t + Mod[t, 2]][3/2 b[n-1]]];
a[n_] := b[n+1] - 3/2 b[n] + 1;
Array[a, 120] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz *)

Formula

a(n) = A304273(n+1) + 1.

From Alois P. Heinz, Jun 21 2018: (Start)

a(n) = A305498(n+1) -3/2*A305498(n) + 1.

Sum_{i=0..n-1} (3/2)^i*a(n-i) = A305497(n). (End)

Extensions

More terms from Alois P. Heinz, Jun 21 2018