This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A222813 #27 Feb 21 2024 08:19:41
%S A222813 3,7,15,31,51,63,99,127,195,231,255,387,455,511,771,819,903,975,1023,
%T A222813 1539,1651,1799,1935,2047,3075,3171,3315,3591,3687,3855,3999,4095,
%U A222813 6147,6371,6643,7175,7399,7695,7967,8191,12291,12483,12771,13107,13299,14343,14535,14823,15375,15567,15903,16191,16383,24579
%N A222813 Numbers whose binary representation is palindromic and in which all runs of 0's and 1's have length at least 2.
%C A222813 These are the decimal representations of A061851 read as base-2 numbers.
%C A222813 The terms with an odd number L = 2k-1 of bits, i.e., 2^(L-1) < a(n) < 2^L, are given by the terms of A033015 with length k, shifted k-1 digits to the left and 'OR'ed with the binary reversal of the term. Terms with an even number L = 2k of digits are given as m*2^k + (binary reversal of m) where m runs over the k-bit terms from A033015 and the k-1 bit terms with the last bit negated appended). This explains the FORMULA for the number of terms of given size. - _M. F. Hasler_, Oct 17 2022
%H A222813 Ray Chandler, <a href="/A222813/b222813.txt">Table of n, a(n) for n = 1..10000</a> (first 608 terms from N. J. A. Sloane)
%F A222813 From _M. F. Hasler_, Oct 06 2022: (Start)
%F A222813 Intersection of A006995 and A033015: binary palindromes with no isolated digit.
%F A222813 There are A000045(A004526(k)) = Fibonacci(floor(k/2)) terms between 2^(k-1) and 2^k.
%F A222813 a(n) = A028897(A061851(n)), where A028897 = convert binary to decimal. (End)
%e A222813 51 (base 10) = 110011 (base 2), which is a palindrome and has three runs all of length 2.
%t A222813 brpalQ[n_]:=Module[{idn2=IntegerDigits[n,2]},idn2==Reverse[idn2] && Min[ Length/@ Split[idn2]]>1]; Select[Range[25000],brpalQ] (* _Harvey P. Dale_, May 21 2014 *)
%o A222813 (PARI) is(n)=is_A033015(n)&&Vecrev(n=binary(n))==n \\ _M. F. Hasler_, Oct 06 2022
%o A222813 (PARI) {A222813_row(n, s=A033015_row(n\/2))=apply(A030101, if(n%2, s\2, n>2, s=setunion([k*2+1-k%2|k<-A033015_row(n\2-1)],s), s=[1]))+s<<(n\2)} \\ Terms with n bits, i.e. between 2^(n-1) and 2^n. - _M. F. Hasler_, Oct 17 2022
%Y A222813 Cf. A061851.
%Y A222813 Cf. A006995 (binary palindromes), A033015 (no isolated binary digit), A028897 ("rebase" 10 -> 2).
%K A222813 nonn,base
%O A222813 1,1
%A A222813 _N. J. A. Sloane_, Mar 11 2013