A145342 -id:A145342 - cp's OEIS frontend

A145341 Convert 2n-1 to binary. Reverse its digits. Convert back to decimal to get a(n).

1, 3, 5, 7, 9, 13, 11, 15, 17, 25, 21, 29, 19, 27, 23, 31, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 65, 97, 81, 113, 73, 105, 89, 121, 69, 101, 85, 117, 77, 109, 93, 125, 67, 99, 83, 115, 75, 107, 91, 123, 71, 103, 87, 119, 79, 111, 95, 127, 129, 193

Offset: 1

Leroy Quet, Oct 08 2008

This sequence is a permutation of the odd positive integers.
From Yosu Yurramendi, Feb 05 2019: (Start)
If the terms (n > 0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...
   1,
   3,  5,
   7,  9, 13, 11,
  15, 17, 25, 21,  29, 19,  27, 23,
  31, 33, 49, 41,  57, 37,  53, 45,  61, 35,  51, 43,  59, 39,  55, 47,
  63, 65, 97, 81, 113, 73, 105, 89, 121, 69, 101, 85, 117, 77, 109, 93, 125, ...
for m > 0,  a(2^(m+1)) = 2*a(2^m) + 1; a(2^m + 1) = a(2^m) + 2; a(2^(m+1) + 2^m) = 2*a(2^(m+1)) - 1,
for m > 0, 0 < k < 2^m, a(2^(m+1) + k) = 2*a(2^m + k) - 1, a(2^(m+1) + 2^m + k) = a(2^(m+1) + k) + 2.
This relationship is enough to reproduce the sequence.
If the terms (n > 0) are written as an array (right-aligned fashion):
                                                                              1,
                                                                          3,  5,
                                                                 7,  9,  13, 11,
                                              15, 17,  25, 21,  29, 19,  27, 23,
           31, 33, 49, 41,  57, 37,  53, 45,  61, 35,  51, 43,  59, 39,  55, 47,
  ... 93, 125, 67, 99, 83, 115, 75, 107, 91, 123, 71, 103, 87, 119, 79, 111, 95,
...
for m >= 0, a(2^(m+1)+2^m) = 4*a(2^m) + 1.
for m >= 0, 0 <= k < 2^m-1, a(2^(m+2)-1-k) = 2*a(2^(m+1)-1-k) + 1.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A030101, A145342.

Table[FromDigits[Reverse[IntegerDigits[2*n - 1, 2]], 2], {n, 1, 100}] (* Stefan Steinerberger, Oct 11 2008 *)

a(n) = fromdigits(Vecrev(binary(2*n-1)), 2); \\ Michel Marcus, Feb 04 2019

nmax <- 10^3 # by choice
  a <- vector()
  for (o in seq(1,nmax,2)){
    w <- which(as.numeric(intToBits(o))==1)
    a <- c(a, sum(2^(max(w)-w)))
}
a[1:66]
# Yosu Yurramendi, Feb 04 2019

a(n) = A030101(2n-1).

a(n) = A145342(n)*2 - 1.

More terms from R. J. Mathar, Ray Chandler and Stefan Steinerberger, Oct 10 2008

A153036 Integer parts of the full Stern-Brocot tree.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Dec 22 2008

			a(1): 1;
a(2..3): 1x0, 2;
a(4..7): 2x0, 1x1, 3;
a(8..15): 4x0, 2x1, 1x2, 4;
a(16..31): 8x0, 4x1, 2x2, 1x3, 5;
a(32..63): 16x0, 8x1, 4x2, 2x3, 1x4, 6;
a(64..127): 32x0, 16x1, 8x2, 4x3, 2x4, 1x5, 7;
a(128..255): 64x0, 32x1, 16x2, 8x3, 4x4, 2x5, 1x6, 8;
a(256..511): 128x0, 64x1, 32x2, 16x3, 8x4, 4x5, 2x6, 1x7, 9.

Jon Maiga, Computer-generated formulas for A153036, Sequence Machine.
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences

Cf. A130321.
If every block of terms of length 2^k is reversed, we get A290256; other permutations within these blocks give A007814 and A272729-1.
Cf. A090996, A043545, A007814, A145342.

a(n+1) = floor(A007305(n+2)/A047679(n)). [Corrected by Andrey Zabolotskiy, Jul 23 2020]
a(n) = if n=2^k-1 then k else Log2(n)-1-Log2(2^(Log2(n)+1)-(n+1)), where Log2=A000523.
From Andrey Zabolotskiy, Oct 07 2021: (Start)
Formulas discovered by Sequence Machine (and also essentially by Kevin Ryde):
a(n) = A090996(n) - A043545(n).
a(n) = A007814(A145342(n+1)). (End)

a(0) = 0 added by Andrey Zabolotskiy, Jul 23 2020

cp's OEIS Frontend

A145341 Convert 2n-1 to binary. Reverse its digits. Convert back to decimal to get a(n).

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