A371032 -id:A371032 - cp's OEIS frontend

A065447 Concatenation of 1, 00, 111, 0000, ..., n 1's (if n is odd) or n 0's (if n is even).

1, 100, 100111, 1001110000, 100111000011111, 100111000011111000000, 1001110000111110000001111111, 100111000011111000000111111100000000, 100111000011111000000111111100000000111111111, 1001110000111110000001111111000000001111111110000000000

Offset: 1

Lior Manor, Nov 18 2001

a(n) is divisible by A002275([(n+1)/2]) = (10^[(n+1)/2]-1)/9. Cf. A262806. - Max Alekseyev, Jun 02 2013

The unique sequence of binary words a(n) such that the k-th run of a(n) has length k, for k = 1..n . - Clark Kimberling, Mar 08 2024

			a(2) = 100, the concatenation of one 1, two 0's.
a(3) = 100111, the concatenation of one 1, two 0's, three 1's.
a(4) = 1001110000, the concatenation of one 1, two 0's, three 1's, four 0's.

Harry J. Smith, Table of n, a(n) for n = 1..44

For decimal version see A065760.

Cf. A000217, A065761, A371032.

a:= n-> parse(cat((irem(i,2)$i)$i=1..n)):
seq(a(n), n=1..10);  # Alois P. Heinz, Mar 08 2024

FoldList[Join, {1}, Map[ConstantArray[Mod[#, 2], #] &, Range[2, 10]]] (* Peter J. C. Moses, Mar 08 2024 *)

{ m=10; for (n=1, 44, if (n==1, a=1, m*=10; a*=m; if (n%2, a+=(m - 1)/9)); write("b065447.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 19 2009

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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 18 2024

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

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

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

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

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

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

New name from Michel Marcus, Jul 09 2024

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

cp's OEIS Frontend

A065447 Concatenation of 1, 00, 111, 0000, ..., n 1's (if n is odd) or n 0's (if n is even).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI