A074330 a(n) = Sum_{k=1..n} 2^b(k) where b(k) denotes the number of 1's in the binary representation of k.

2, 4, 8, 10, 14, 18, 26, 28, 32, 36, 44, 48, 56, 64, 80, 82, 86, 90, 98, 102, 110, 118, 134, 138, 146, 154, 170, 178, 194, 210, 242, 244, 248, 252, 260, 264, 272, 280, 296, 300, 308, 316, 332, 340, 356, 372, 404, 408, 416, 424, 440, 448, 464, 480, 512, 520, 536

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 29.

Formula

a(n+1)-a(n) = A001316(n)

From Ralf Stephan, Oct 07 2003: (Start)

a(0)=0, a(2n) = 2a(n-1) + a(n) + 2, a(2n+1) = 3a(n) + 2.

G.f.: (1/(1-x)) * Product_{k>=0} (1 + 2x^2^k). (End)

Maple

f:= proc(n) option remember; if n::even then 2*procname(n/2-1)+procname(n/2)+2
  else 3*procname((n-1)/2)+2
  fi
end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Oct 08 2020

Mathematica

b[n_] := IntegerDigits[n, 2] // Total;
a[n_] := 2^(b /@ Range[n]) // Total;
Array[a, 100] (* Jean-François Alcover, Aug 16 2022 *)

Prog

(PARI) a(n)=sum(i=1, n, 2^sum(k=1, length(binary(i)), component(binary(i), k)))

Crossrefs

a(n) = A006046(n+1)-1. Cf. A080263.

Sequence in context: A300781 A050567 A069879 * A317626 A292550 A024895

Adjacent sequences: A074327 A074328 A074329 * A074331 A074332 A074333

Keyword

easy,nonn

Author

Benoit Cloitre, Oct 06 2002

Status

approved