A230892 - OEIS

%I #41 Mar 02 2019 02:52:03

%S 0,3,1,2,4,7,8,5,6,9,10,12,15,16,11,13,14,17,18,20,23,24,27,29,30,32,

%T 19,21,22,25,26,28,31,33,34,36,39,40,43,45,46,48,51,53,54,57,58,60,63,

%U 64,35,37,38,41,42,44,47,49,50,52,55,56,59,61,62,65,66,68,71

%N Terms of A230891 written in base 10: the binary expansions of a(n) and a(n+1) taken together can be rearranged to form a palindrome.

%C See A230891 for precise definition.

%C Just as for A228407, we can ask: does every number appear? The answer is yes - see the Comments in A228407.

%C The difference d(n)=a(n)-n increases from d(3*2^(k-2)+2) = 1-2^(k-2) to d(3*2^(k-1)+1) = 1-2^(k-1), going through 0 at n=2^k+1 and n=2^k+2, cf. examples. - _M. F. Hasler_, Nov 12 2013

%C From _Robert G. Wilson v_, Nov 15 2013: (Start)

%C Beginning with k=3, each "grouping" of ever increasing terms, begins at 2^k + 3 and runs up to 2^(k+2) and includes 3*2^(k-1) terms.

%C Indices of powers of 2 occur at: 2, 3, 4, 6, 13, 25, 49, 97, 193, 385, 769, 1537, ..., which, except for 2, 3 & 6, is A181565: 3*2^n + 1.

%C When the index equals the term: 0, 4, 9, 10, 17, 18, 33, 34, 65, 66, 129, 130, 257, 258, 513, 514, 1025, 1026, 2049, 2050, ..., .

%C Parity of a(n) beginning at n=0: 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ..., . (End)

%H Chai Wah Wu, <a href="/A230892/b230892.txt">Table of n, a(n) for n = 0..30000</a> (first 2051 terms from Robert G. Wilson v)

%H Robert G. Wilson v, <a href="/A230892/a230892.pdf">Graph of the first 1025 terms</a>

%e From _M. F. Hasler_, Nov 12 2013: (Start)

%e Let d(n)=a(n)-n, i.e., a(n)=n+d(n). Then we have, after initial values d(0..8)=(0, 2, -1, -1, 0, 2, 2, -2, -2), the

%e following pattern: d(9) = d(10) = 0, ..., d(13) = 3,

%e d(14) = -3, ..., d(17) = d(18) = 0, ..., d(25) = 7;

%e d(26) = -7, ..., d(33) = d(34) = 0, ..., d(49) = 15,

%e d(50) = -15, ..., d(65) = d(66) = 0, ..., d(97) = 31,

%e d(98) = -31, ..., d(129) = d(130) = 0, ..., d(193) = 63,

%e d(194) = -63,..., d(257) = d(258) = 0, ... (End)

%o (PARI) {u=0; a=0; La=1; ha=0/*hack*/; for(n=1, 99, u+=1<<a; print1(a", "); L=1; for(k=1, 9e9, bittest(u, k)&&next; while(k>=2^L,L++); bittest(ha+h=hammingweight(k),0)&&!bittest(La+L,0)&&next; !a&&k<3&&next; a=k; ha=h; La=L; break))} \\ _M. F. Hasler_, Nov 11 2013

%o (Python)

%o from collections import Counter

%o A230892_list, l, s, b = [0, 3], Counter('11'), 1, {3}

%o for _ in range(30001):

%o     i = s

%o     while True:

%o         if i not in b:

%o             li, o = Counter(bin(i)[2:]), 0

%o             for d in (l+li).values():

%o                 if d % 2:

%o                     if o > 0:

%o                         break

%o                     o += 1

%o             else:

%o                 A230892_list.append(i)

%o                 l = li

%o                 b.add(i)

%o                 while s in b:

%o                     b.remove(s)

%o                     s += 1

%o                 break

%o         i += 1 # _Chai Wah Wu_, Jun 19 2016

%Y Cf. A230891, A228407.

%K nonn,base

%O 0,2

%A _N. J. A. Sloane_, Nov 11 2013