A127804 a(2n) = 2^(2n), a(2n+1) = 2^(2n+1) + a(n).

1, 3, 4, 11, 16, 36, 64, 139, 256, 528, 1024, 2084, 4096, 8256, 16384, 32907, 65536, 131328, 262144, 524816, 1048576, 2098176, 4194304, 8390692, 16777216, 33558528, 67108864, 134225984, 268435456, 536887296, 1073741824, 2147516555, 4294967296, 8590000128

Offset: 0

Author

Paul Barry, Jan 29 2007

Keywords

• nonn
• easy

Comments

From Tilman Piesk, Jun 30 2025: (Start)

The binary expansion of a(n), with bits least to most significant, is row n of A115361.

Number of 1's in binary expansion of a(n) is A001511(n+1).

Row sums of triangle A128807.

Row sums of triangle A127803.(End)

Links

• Tilman Piesk, Table of n, a(n) for n = 0..1023 (terms 0..61 from R. J. Mathar)
• Tilman Piesk, Illustrated triangles with 32 rows:
• related to necklaces (similar to A385665),
• binary (A115361),
• signed (A127803)
• Tilman Piesk, Serration of Boolean functions (Wikiversity)

Crossrefs

Cf. A045654, A128807, A127803, A115361, A187767, A001511.

Programs

Maple

A127804 := proc(n)
    add( A127803(n,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 12 2024
# second Maple program:
a:= proc(n) option remember;
      2^n+`if`(n::odd, a((n-1)/2), 0)
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jul 01 2025

Mathematica

rows = 30;
A[n_, k_] := If[k <= n, If[n <= 2 k, 1/(2*2^n - 1), 0], 0];
T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
a[n_] := T[[n+1]] // Total;
Table[a[n], {n, 0, rows-1}] (* Jean-François Alcover, Jul 03 2018 *)

Formula

Conjecture: a(n) = 1 + A187767(n+1). - Andrew Howroyd, Jun 02 2017

From Tilman Piesk, Jun 30 2025: (Start)

a(n) = Sum_{i=0..A001511(n+1)-1} 2^((n+1) / 2^i - 1)

= Sum_{i=0..A001511(n+1)-1} 2^floor(n / 2^i).

a(n) = A045654(n+1) / 2. (End)

Extensions

Name changed by Tilman Piesk, Jun 30 2025