A037834 a(n) = Sum_{i=1..m} |d(i) - d(i-1)|, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 3, 4, 5

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

Number of i such that |d(i) - d(i-1)| = 1, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n)+1 is the number of iterations of the map x -> A035327(x) needed to reach 0 (see A005811(n) for n>=1). - Flávio V. Fernandes, Apr 28 2025

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A005811, A030308, A035327.

Programs

Haskell

a037834 n = sum $ map fromEnum $ zipWith (/=) (tail bs) bs
            where bs = a030308_row n
-- Reinhard Zumkeller, Feb 20 2014

Maple

A037834 := proc(n)
    local dgs ;
    dgs := convert(n,base,2);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc: # R. J. Mathar, Oct 16 2015

Mathematica

Table[Total@ Flatten@ Map[Abs@ Differences@ # &, Partition[ IntegerDigits[n, 2], 2, 1]], {n, 90}] (* Michael De Vlieger, May 09 2017 *)

Python

def A037834(n): return (n^(n>>1)).bit_count()-1 # Chai Wah Wu, Jul 13 2024

Formula

a(n) = A005811(n)-1.