A298011 If n = Sum_{i=1..h} 2^b_i with 0 <= b_1 < ... < b_h, then a(n) = Sum_{i=1..h} i * 2^b_i.

0, 1, 2, 5, 4, 9, 10, 17, 8, 17, 18, 29, 20, 33, 34, 49, 16, 33, 34, 53, 36, 57, 58, 81, 40, 65, 66, 93, 68, 97, 98, 129, 32, 65, 66, 101, 68, 105, 106, 145, 72, 113, 114, 157, 116, 161, 162, 209, 80, 129, 130, 181, 132, 185, 186, 241, 136, 193, 194, 253, 196

Offset: 0

Author

Rémy Sigrist, Jan 10 2018

Keywords

• nonn
• base
• easy

Comments

This sequence is similar to A298043.

Examples

			For n = 42:
- 42 = 2 + 8 + 32,
- hence a(42) = 1*2 + 2*8 + 3*32 = 114.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Colored scatterplot of a(n) for n = 0..2^16 (where the color is function of A000120(n)).

Crossrefs

Cf. A000120, A000337, A129760, A298043.

Programs

Maple

F[0]:= x -> x:
for i from 1 to 8 do
  F[i]:= unapply(convert(series(2*(x+1)*F[i-1](x^2)+H, x, 2^(i+1)),
    polynom), x)
od:
seq(coeff(F[8](x),x,j),j=0..2^9-1); # Robert Israel, Jan 16 2018

Mathematica

a[0] = 0; a[n_] := a[n] = If[OddQ[n], a[n - 1] + n, 2*a[n/2]]; Array[a, 100, 0] (* Amiram Eldar, Jul 24 2023 *)

PARI

a(n) = my (b=binary(n), z=0); forstep (i=#b, 1, -1, if (b[i], b[i] = z++)); return (fromdigits(b, 2))

PARI

first(n) = n += (n-1)%2; my(res = vector(n)); res[1]= 1; for(i = 1, n\2, res[2 * i] = 2 * res[i]; res[2 * i + 1] = res[2 * i] + 2*i + 1); concat([0], res) \\ David A. Corneth, Jan 14 2018

Formula

a(n) = Sum_{k = 0..A000120(n)-1} A129760^k(n) for any n > 0 (where A129760^k denotes the k-th iterate of A129760).

a(n) >= n with equality iff n = 0 or n = 2^k for some k >= 0.

a(2 * n) = 2 * a(n).

a(2^n - 1) = A000337(n).

a(2 * n + 1) = a(2 * n) + 2 * n + 1. David A. Corneth, Jan 14 2018

G.f. g(x) satisfies g(x) = 2*(x+1)*g(x^2) + x*(1+x^2)/(1-x^2)^2. - Robert Israel, Jan 16 2018