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

Author

Leroy Quet, Oct 08 2008

Keywords

• base
• nonn

Comments

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)

Links

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

Crossrefs

Cf. A030101, A145342.

Programs

Mathematica

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

PARI

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

R

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

Formula

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

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

Extensions

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