A361074 Sum of the j-th number with binary weight n-j+1 over all j in [n].

0, 1, 5, 16, 40, 92, 193, 401, 812, 1632, 3261, 6526, 13030, 26049, 52013, 103974, 207797, 415496, 830636, 1661086, 3321498, 6642591, 13283920, 26567121, 53131653, 106261922, 212518857, 425034976, 850060303, 1700115399, 3400211408, 6800412866, 13600787296

Offset: 0

Author

Alois P. Heinz, Mar 01 2023

Keywords

• nonn
• base

Examples

			a(0) = 0 (empty sum).
a(1) = 1 = 1_2.
a(2) = 5 = 2 + 3 = 10_2 + 11_2.
a(3) = 16 = 4 + 5 + 7 = 100_2 + 101_2 + 111_2.
a(4) = 40 = 8 + 6 + 11 + 15 = 1000_2 + 110_2 + 1011_2 + 1111_2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Antidiagonal sums of A066884 or of A067576.

Cf. A000120, A057168.

Programs

Maple

b:= proc(i, j) option remember; uses Bits: local c, l, k;
      if j=1 then 2^i-1
    else c, l:= 0, [Split(b(i, j-1))[], 0];
         for k while l[k]<>1 or l[k+1]<>0 do c:=c+l[k] od;
         Join([1$c, 0$k-c, 1, l[k+2..-1][]])
      fi
    end:
a:= n-> add(b(j, n-j+1), j=1..n):
seq(a(n), n=0..32);

Formula

a(n) = Sum_{j=1..n} A066884(j,n-j+1) = Sum_{j=1..n} A067576(j,n-j+1).

Conjecture: a(n) ~ 19 * 2^n / 6. - Vaclav Kotesovec, Mar 04 2023