

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 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 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



FORMULA

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.


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) {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



KEYWORD

nonn,base


AUTHOR



STATUS

approved



