A091311 Partial sums of 3^A007814(n).

0, 1, 4, 5, 14, 15, 18, 19, 46, 47, 50, 51, 60, 61, 64, 65, 146, 147, 150, 151, 160, 161, 164, 165, 192, 193, 196, 197, 206, 207, 210, 211, 454, 455, 458, 459, 468, 469, 472, 473, 500, 501, 504, 505, 514, 515, 518, 519, 600, 601, 604, 605, 614, 615, 618, 619

Offset

0,3

Links

Table of n, a(n) for n=0..55.

Formula

Recurrence: a(2n) = 3*a(n) + n, a(2n+1) = 3*a(n) + n + 1.

G.f.: 1/(1-x) * Sum_{k>=0} 3^k*t/(1-t^2), where t = x^2^k.

a(n) = 2*A005836(n+1) - n.

a(n) = n + 2*Sum_{k>=1} 3^(k-1)*floor(n/2^k). - Ridouane Oudra, May 16 2026

Mathematica

Join[{0}, Accumulate[3^IntegerExponent[Range[64], 2]]] (* Harvey P. Dale, May 20 2025 *)

Prog

(PARI) a(n)=sum(k=1, n, 3^valuation(k, 2))
(PARI) a(n)=if(n<1, 0, if(n%2==0, 3*a(n/2)+n/2, 3*a((n-1)/2)+(n+1)/2))
(Python)
def A091311(n): return (int(bin(n)[2:], 3)<<1)-n # Chai Wah Wu, Jul 07 2022

Crossrefs

First differences are A061393(n)-1.

Cf. A005836, A007814, A006520, A224915.

Sequence in context: A195747 A238315 A392421 * A008540 A290810 A000867

Adjacent sequences: A091308 A091309 A091310 * A091312 A091313 A091314

Keyword

nonn,easy

Author

Ralf Stephan, Feb 24 2004

Status

approved