A161400 Positive integers that are palindromes (of even length) in binary, each made by concatenating two identical binary palindromes.

3, 15, 45, 63, 153, 255, 561, 693, 891, 1023, 2145, 2925, 3315, 4095, 8385, 9417, 10965, 11997, 12771, 13803, 15351, 16383, 33153, 39321, 42405, 48573, 50115, 56283, 59367, 65535, 131841, 140049, 152361, 160569, 166725, 174933, 187245

Offset

1,1

Comments

If m is the n-th positive integer that is a binary palindrome, and m written in binary is k digits long, then a(n) = m*(2^k +1).

Links

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Formula

a(n) = (2^A070939(p)+1)*p where p = A006995(n+1). [From R. J. Mathar, Sep 27 2009]

Example

The first eight terms of this sequence written in binary: 11, 1111, 101101, 111111, 10011001, 11111111, 1000110001, 1010110101.

Mathematica

Union[Flatten[Table[FromDigits[Join[#, #], 2]&/@Select[Tuples[ {1, 0}, n], First[ #]!=0&&Last[#]!=0&&#==Reverse[#]&], {n, 10}]]] (* Harvey P. Dale, Jul 15 2014 *)

Prog

(Python)
from itertools import product
def bin_pals():
  yield "1"
  digits, midrange = 2, [[""], ["0", "1"]]
  while True:
    for p in product("01", repeat=digits//2-1):
      left = "1"+"".join(p)
      for middle in midrange[digits%2]:
        yield left+middle+left[::-1]
    digits += 1
def aupton(terms):
  alst, bgen = [], bin_pals()
  while len(alst) < terms: b = next(bgen); alst.append(int(b+b, 2))
  return alst
print(aupton(37)) # Michael S. Branicky, Mar 15 2021

Crossrefs

Cf. A006995.

Sequence in context: A127407 A196237 A177146 * A112810 A334078 A094191

Adjacent sequences: A161397 A161398 A161399 * A161401 A161402 A161403

Keyword

base,nonn

Author

Leroy Quet, Jun 09 2009

Extensions

Extended beyond 693 by R. J. Mathar, Sep 27 2009

Status

approved