A371033 a(n) is the integer whose binary expansion starts with 1 and such that the runs of identical bits have lengths n, n-1, n-2, ..., 3, 2, 1.

1, 6, 57, 966, 31801, 2065350, 266370105, 68453106630, 35115918982201, 35993681099981766, 73750982613738224697, 302157703921043555451846, 2475577920866839506242796601, 40562343629382474008388259775430, 1329187433441286490429798672020569145

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

a(n) == n (mod 2). - Alois P. Heinz, Jul 09 2024

a(n) = 2^(n*(n+1)/2) - 1 - a(n-1). - Robert Israel, Jul 09 2024

Example

Representations as binary words (as in A371032) have decreasing runlengths:
    1:  1
    6:  110
   57:  111001
  966:  1111000110  (runlengths 4,3,2,1)

Maple

a:= n-> Bits[Join]([seq((1-(n-i) mod 2)$i, i=1..n)]):
seq(a(n), n=1..15);  # Alois P. Heinz, Jul 09 2024

Mathematica

Map[FromDigits[#, 2] &, Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 16}]]    (* Peter J. C. Moses, Mar 08 2024 *)

Prog

(Python)
def A371033(n):
    c = 0
    for i in range(n):
        c <<= n-i
        if i&1^1:
            c += (1<<n-i)-1
    return c # Chai Wah Wu, Mar 18 2024

Crossrefs

Cf. A006125, A007088, A065760, A126883, A371032 (binary version).

Sequence in context: A361291 A145170 A180255 * A034218 A034237 A291628

Adjacent sequences: A371030 A371031 A371032 * A371034 A371035 A371036

Keyword

nonn,base,easy

Author

Clark Kimberling, Mar 18 2024

Extensions

New name from Michel Marcus, Jul 09 2024

a(15) corrected by Alois P. Heinz, Jul 09 2024

Status

approved