A162751 Write down in binary the n-th positive (odd) integer that is a palindrome in base 2. Take only the leftmost half of the digits (including the middle digit if there are an odd number of digits). a(n) is the decimal equivalent of the result.

1, 1, 2, 3, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44

Offset: 1

Author

Leroy Quet, Jul 12 2009

Keywords

• base
• nonn

Comments

Every positive integer occurs exactly twice in this sequence.

Examples

			27 is the 9th (odd) palindrome when written in binary. 27 in binary is 11011. Take the leftmost half of the digits (including the middle digit), and we have 110. a(9) is decimal equivalent of this, which is 6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006995.

Programs

Maple

read("transforms3") ; a006995 := BFILETOLIST("b006995.txt") ; chop := proc(L) [op(1.. floor((nops(L)+1)/2),L)] ; end: for n from 2 to 100 do p := op(n,a006995) ; bdgs := chop(convert(p,base,2)) ; add(op(-i,bdgs)*2^(i-1),i=1..nops(bdgs)) ; printf("%d,",%) ; end do: # R. J. Mathar, Aug 01 2009
A162751:= proc(n) option remember;
          if n <= 2 then 1
          elif n::odd then 2*procname((n-1)/2)
          else 2*procname(n/2-1)+1
          end if
end proc;  # Robert Israel, Apr 03 2014

Mathematica

a[n_] := a[n] = If[n <= 2, 1, If[OddQ[n], 2 a[(n-1)/2], 2 a[n/2-1] + 1]];
Array[a, 75] (* Jean-François Alcover, Apr 06 2020, after Robert Israel *)

Formula

a(1) = a(2) = 1; for i >= 2, a(2 i-1) = 2 a(i-1) and a(2 i) = 2 a(i-1) + 1. Robert Israel, Apr 03 2014

Extensions

More terms from R. J. Mathar, Aug 01 2009