cp's OEIS Frontend

A061851 Digital representation of n contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's.

%I A061851 #33 Jan 09 2025 10:16:00
%S A061851 11,111,1111,11111,110011,111111,1100011,1111111,11000011,11100111,
%T A061851 11111111,110000011,111000111,111111111,1100000011,1100110011,
%U A061851 1110000111,1111001111,1111111111,11000000011,11001110011,11100000111,11110001111,11111111111,110000000011,110001100011
%N A061851 Digital representation of n contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's.
%C A061851 The terms can be constructed by gluing together terms from A355280 with their reversal as follows: The terms with odd length L = 2k-1 are given from the k-digit terms of A355280 by replacing the last digit with the reversal of the term. (Equivalently, concatenate with the reversal and delete one of the middle digits.) Terms with an even number L = 2k of digits are given as concatenation(m, reverse(m)) = m*10^L(m) + A004086(m) where m runs over the k-digit terms from A355280, and the (k-1)-digit terms with the 1's complement of the last digit appended. This explains the formula given in CROSSREFS for the number of terms of given length. - _M. F. Hasler_, Oct 17 2022
%H A061851 Ray Chandler, <a href="/A061851/b061851.txt">Table of n, a(n) for n = 1..10000</a> (first 608 terms from N. J. A. Sloane)
%H A061851 N. J. A. Sloane, <a href="http://arxiv.org/abs/1307.0453">2178 And All That</a>, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
%H A061851 N. J. A. Sloane, <a href="/A001232/a001232.pdf">2178 And All That</a> [Local copy]
%F A061851 a(n) = A001232(n)/99 = A008918(n)/198.
%F A061851 a(n) = A007088(A222813(n)), where A007088 = write in binary. - _M. F. Hasler_, Oct 06 2022
%o A061851 (PARI) concat(apply( {A061851_row(n)=[fromdigits(binary(m))|m<-A222813_row(n)]}, [1..11])) \\ _M. F. Hasler_, Oct 17 2022
%o A061851 (Python)
%o A061851 def A061851_row(n): return [] if n < 2 else [10**n//9] if n < 6 else [
%o A061851     m*10**(n//2) + A004086(m//10) for m in A355280_row(n//2+1)] if n&1 else [
%o A061851     m*10**(n//2) + A004086(m) for m in sorted(A355280_row(n//2)+
%o A061851                         [x*10+1-x%10 for x in A355280_row(n//2-1)])]
%o A061851 # _M. F. Hasler_, Oct 17 2022
%Y A061851 Number of terms with k digits is Fibonacci(floor(k/2)) = A000045(A004526(k)).
%Y A061851 Union of this sequence and twice this sequence is A061852 and 99 times that is A008919.
%Y A061851 Cf. A214927, A001232, A222814, A008918, A008919, A222813 (terms converted to decimal).
%Y A061851 Cf. A355280 (palindromic binary numbers with no digit run of length < 2).
%K A061851 base,nonn
%O A061851 1,1
%A A061851 _Henry Bottomley_, May 10 2001