A264964 Numbers that are the sum of two binary palindromes of the same (binary) length.

0, 2, 6, 10, 12, 14, 18, 24, 30, 34, 38, 42, 44, 48, 52, 54, 58, 62, 66, 78, 84, 90, 96, 102, 108, 114, 126, 130, 138, 146, 150, 158, 164, 166, 170, 172, 178, 180, 184, 186, 192, 198, 200, 204, 206, 212, 214, 218, 220, 226, 234, 238, 246, 254, 258, 282, 294, 306, 318, 324, 330, 342, 348, 354, 360, 372, 378, 384, 390, 396

Offset

1,2

Comments

Theorem: Adding two binary palindromes of length k >= 2 in all possible ways produces 3^floor((k-1)/2) distinct sums. (There are 2^floor((k-1)/2) binary palindromes of length k - see A006995.)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..13122

Example

There are four binary palindromes of length 5, namely (written in base 10) 17, 21, 27, 31, and adding them in pairs gives nine distinct numbers: 34, 38, 42, 44, 48, 52, 54, 58, 62.
There are eight binary palindromes of length 7, namely (written in base 10) 65, 73, 85, 93, 99, 107, 119, 127, and adding them in pairs gives 27 distinct numbers: 130, 138, 146, 150, 158, 164, 166, 170, 172, 178, 180, 184, 186, 192, 198, 200, 204, 206, 212, 214, 218, 220, 226, 234, 238, 246, 254.

Mathematica

f[n_] := Select[Map[FromDigits /@ IntegerDigits[#, 2] &, Map[Function[k, {k, # - k}], Range@ Floor[#/2]] &@ n], AllTrue[#, Reverse@ # == # &@ IntegerDigits@ # &] && IntegerLength@ First@ # == IntegerLength@ Last@ # &]; Prepend[Select[Range@ 400, Length@ f@ # > 0 &], 0] (* Michael De Vlieger, Nov 29 2015, Mma version 10 *)
Join[{0}, Table[Total/@Tuples[FromDigits[#, 2]&/@Select[Tuples[{1, 0}, n], #[[1]] != 0&&#==Reverse[#]&], 2]//Union, {n, 8}]//Flatten] (* Harvey P. Dale, Apr 12 2017 *)

Crossrefs

Cf. A006995, A261679, A241491, A261678.

Sequence in context: A057921 A095300 A097381 * A195064 A328926 A055743

Adjacent sequences: A264961 A264962 A264963 * A264965 A264966 A264967

Keyword

nonn,base

Author

N. J. A. Sloane, Nov 29 2015

Status

approved