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

%I A264964 #28 Apr 12 2017 04:43:19
%S A264964 0,2,6,10,12,14,18,24,30,34,38,42,44,48,52,54,58,62,66,78,84,90,96,
%T A264964 102,108,114,126,130,138,146,150,158,164,166,170,172,178,180,184,186,
%U A264964 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
%N A264964 Numbers that are the sum of two binary palindromes of the same (binary) length.
%C A264964 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.)
%H A264964 Chai Wah Wu, <a href="/A264964/b264964.txt">Table of n, a(n) for n = 1..13122</a>
%e A264964 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.
%e A264964 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.
%t A264964 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 *)
%t A264964 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 *)
%Y A264964 Cf. A006995, A261679, A241491, A261678.
%K A264964 nonn,base
%O A264964 1,2
%A A264964 _N. J. A. Sloane_, Nov 29 2015