OFFSET
1,2
COMMENTS
Note: you get A006995 (all binary palindromes) if you take (after zero) alternatively 2^n (starting from 2^0 = 1) terms from A048700 and as many from A048701 and then each time, twice as many from both.
A178225(a(n)) = 1. - Reinhard Zumkeller, Oct 21 2011
Comment from Altug Alkan, Dec 03 2015: (Start)
a(6*k) is divisible by 9 for k > 0.
a(3*k+(-1)^k-2) is divisible by 3 for k > 1.
The minimum value of a(n+1) - a(n) occurs when n = 2.
A014551(n) appears in this sequence for n > 0. (End)
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (2^(floor_log_2(n)))*n + sum('(bit_i(n, i)*(2^(floor_log_2(n)-i)))', 'i'=1..floor_log_2(n));
a(A047264(n)) mod 3 = 0, for n > 1. - Altug Alkan, Dec 03 2015
MAPLE
bit_i := (x, i) -> `mod`(floor(x/(2^i)), 2);
floor_log_2 := proc(n) local nn, i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi: nn := floor(nn/2); od: end:
MATHEMATICA
Select[Range@ 1651, # == Reverse@ # && OddQ@ Length@ # &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Dec 03 2015 *)
PROG
(PARI) {a(n) = local(f); if( n<1, 0, f = length(binary(n)) - 1; 2^f*n + sum(i=1, f, bittest(n, i) * 2^(f-i)))}; /* Michael Somos, Nov 27 2002 */
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
import Data.List (unfoldr)
a048700 n = a048700_list !! (n-1)
a048700_list = f 1 $ singleton 1 where
f z s = m : f (z+1) (insert (c 0) (insert (c 1) s')) where
c d = foldl (\v d -> 2 * v + d) 0 $ (reverse b) ++ [d] ++ b
b = unfoldr
(\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) z
(m, s') = deleteFindMin s
-- Reinhard Zumkeller, Oct 21 2011
(Python)
def A048700(n):
s = bin(n)[2:]
return int(s+s[-2::-1], 2) # Chai Wah Wu, Feb 26 2021
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Mar 07 1999
STATUS
approved