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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These are the decimal representations of A061851 read as base-2 numbers.
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
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^(k-1) 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+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
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
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Mar 11 2013
STATUS
approved

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Last modified June 13 10:09 EDT 2024. Contains 373383 sequences. (Running on oeis4.)