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, 4, 3, 4, 3, 2, 1, 2, 3, 2, 3

Offset

1,5

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

a(n) is the number of bit transitions (changes between 0 and 1) in the binary representation of the integer n. - Andres Cicuttin, Apr 26 2026

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

Formula

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

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
# Alternative:
a:= n-> add(i, i=Bits[Split](Bits[Xor](n, iquo(n, 2))))-1:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 27 2026

Mathematica

Table[Total@ Flatten@ Map[Abs@ Differences@ # &, Partition[ IntegerDigits[n, 2], 2, 1]], {n, 90}] (* Michael De Vlieger, May 09 2017 *)
(* Alternative: *)
a[n_] := Total@ Abs@ Differences@ IntegerDigits[n, 2]; (* Andres Cicuttin, Apr 26 2026 *)

Prog

(Haskell)
a037834 n = sum $ map fromEnum $ zipWith (/=) (tail bs) bs
            where bs = a030308_row n
-- Reinhard Zumkeller, Feb 20 2014
(Python)
def A037834(n): return (n^(n>>1)).bit_count()-1 # Chai Wah Wu, Jul 13 2024

Crossrefs

Cf. A005811, A030308, A035327.

Sequence in context: A002819 A357562 A307672 * A212496 A380268 A379850

Adjacent sequences: A037831 A037832 A037833 * A037835 A037836 A037837

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved