

A222813


Numbers whose binary representation is palindromic and in which all runs of 0's and 1's have length at least 2.


3



3, 7, 15, 31, 51, 63, 99, 127, 195, 231, 255, 387, 455, 511, 771, 819, 903, 975, 1023, 1539, 1651, 1799, 1935, 2047, 3075, 3171, 3315, 3591, 3687, 3855, 3999, 4095, 6147, 6371, 6643, 7175, 7399, 7695, 7967, 8191, 12291, 12483, 12771, 13107, 13299, 14343, 14535, 14823, 15375, 15567, 15903, 16191, 16383, 24579
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OFFSET

1,1


COMMENTS

These are the decimal representations of A061851 read as base2 numbers.
The terms with an odd number L = 2k1 of bits, i.e., 2^(L1) < a(n) < 2^L, are given by the terms of A033015 with length k, shifted k1 digts 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 kbit terms from A033015 and the k1 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


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 608 terms from N. J. A. Sloane)


FORMULA

From M. F. Hasler, Oct 06 2022: (Start)
Intersection of A006995 and A033015: binary palindromes with no isolated digit.
There are A000045(A004526(k)) = Fibonacci(floor(k/2)) terms between 2^(k1) and 2^k.
a(n) = A028897(A061851(n)), where A028897 = convert binary to decimal. (End)


EXAMPLE

51 (base 10) = 110011 (base 2), which is a palindrome and has three runs all of length 2.


MATHEMATICA

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 *)


PROG

(PARI) is(n)=is_A033015(n)&&Vecrev(n=binary(n))==n \\ M. F. Hasler, Oct 06 2022
(PARI) {A222813_row(n, s=A033015_row(n\/2))=apply(A030101, if(n%2, s\2, n>2, s=setunion([k*2+1k%2k<A033015_row(n\21)], s), s=[1]))+s<<(n\2)} \\ Terms with n bits, i.e. between 2^(n1) and 2^n.  M. F. Hasler, Oct 17 2022


CROSSREFS

Cf. A061851.
Cf. A006995 (binary palindromes), A033015 (no isolated binary digit), A028897 ("rebase" 10 > 2).
Sequence in context: A043729 A331503 A137170 * A304078 A151338 A229006
Adjacent sequences: A222810 A222811 A222812 * A222814 A222815 A222816


KEYWORD

nonn,base


AUTHOR

N. J. A. Sloane, Mar 11 2013


STATUS

approved



