A136013 a(n) = floor(n/2) + 2*a(floor(n/2)), a(0) = 0.

0, 0, 1, 1, 4, 4, 5, 5, 12, 12, 13, 13, 16, 16, 17, 17, 32, 32, 33, 33, 36, 36, 37, 37, 44, 44, 45, 45, 48, 48, 49, 49, 80, 80, 81, 81, 84, 84, 85, 85, 92, 92, 93, 93, 96, 96, 97, 97, 112, 112, 113, 113, 116, 116, 117, 117, 124, 124, 125, 125, 128

Offset

0,5

Comments

A recursive sequence that seems to be related to the ruler function.

It seems that a(2n) = a(2n+1) = A080277(n). - Emeric Deutsch, Mar 31 2008

It appears that if the binary expansion of n is n = Sum b_i*2^i (b_i=0 or 1), then a(n) = Sum i*b_i*2^(i-1). - Marc LeBrun, Sep 07 2015

The observations in the preceding two comments (by Emeric Deutsch and Marc LeBrun) follow from the formulas in A333979. - Pontus von Brömssen, Sep 06 2020

This sequence is a variant of the arithmetic derivative (A003415) based on powers of two instead of primes, because the relation a(m*n) = m*a(n) + n*a(m) holds. If we define the polynomial P(2) = bit0*2^0 + bit1*2^1 + bit2*2^2 + ... = n, and P'(2) is the derivative of P(2), then we will observe P'(2) = a(n). - Thomas Scheuerle, Aug 02 2022

Links

Table of n, a(n) for n=0..60.

Formula

a(n) = A333979(n,2). - Pontus von Brömssen, Sep 06 2020

Maple

a:=proc(n) if n=0 then 0 else floor((1/2)*n)+2*a(floor((1/2)*n)) end if end proc: seq(a(n), n=0..60); # Emeric Deutsch, Mar 31 2008

Mathematica

a = {0}; Do[AppendTo[a, Floor[n/2] + 2*a[[Floor[n/2] + 1]]], {n, 1, 100}]; a (* Stefan Steinerberger, Mar 24 2008 *)
Table[Sum[2^(k-1)*Floor[n*2^-k], {k, 1, Log[2, n]}], {n, 0, 100}] (* Federico Provvedi, Aug 17 2013 *)

Prog

(PARI) a(n) = fromdigits(Vec(Pol(binary(n))'), 2); \\ Kevin Ryde, Apr 29 2021
(Python)
def A136013(n): return sum(map(lambda x:(x[0]+1)*(1<<x[0]), filter(lambda x:x[1]=='1', enumerate(bin(n)[-2:1:-1])))) # Chai Wah Wu, Jul 06 2022

Crossrefs

Cf. A080277, A003415, A333979, A135481 (first differences).

Sequence in context: A152465 A357414 A082968 * A232172 A114884 A132704

Adjacent sequences: A136010 A136011 A136012 * A136014 A136015 A136016

Keyword

nonn,easy

Author

Jack Preston (jpreston(AT)earthlink.net), Mar 20 2008

Extensions

More terms from Stefan Steinerberger and Emeric Deutsch, Mar 24 2008

Spelling corrected by Jason G. Wurtzel, Aug 30 2010

Status

approved