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For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.
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%I #17 Mar 27 2021 11:26:02

%S 0,0,0,1,0,1,3,3,0,0,2,3,6,7,7,7,0,0,0,1,4,5,7,7,12,12,14,15,14,15,15,

%T 15,0,0,0,1,0,1,3,3,8,8,10,11,14,15,15,15,24,24,24,25,28,29,31,31,28,

%U 28,30,31,30,31,31,31,0,0,0,1,0,1,3,3,0,0,2,3,6

%N For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

%C The value of the k-th bit in a(n) corresponds to the most frequent value in the bit triple starting at the k-th bit in n.

%H Rémy Sigrist, <a href="/A342697/b342697.txt">Table of n, a(n) for n = 0..8192</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(n) = 0 iff n belongs to A048715.

%F a(n) = floor(A048730(n)/8) = floor(A048733(n)/2). - _Kevin Ryde_, Mar 26 2021

%e The first terms, in decimal and in binary, are:

%e n a(n) bin(n) bin(a(n))

%e -- ---- ------ ---------

%e 0 0 0 0

%e 1 0 1 0

%e 2 0 10 0

%e 3 1 11 1

%e 4 0 100 0

%e 5 1 101 1

%e 6 3 110 11

%e 7 3 111 11

%e 8 0 1000 0

%e 9 0 1001 0

%e 10 2 1010 10

%e 11 3 1011 11

%e 12 6 1100 110

%e 13 7 1101 111

%e 14 7 1110 111

%e 15 7 1111 111

%o (PARI) a(n) = sum(k=0, #binary(n), ((bittest(n, k)+bittest(n, k+1)+bittest(n, k+2))>=2) * 2^k)

%Y Cf. A048715, A048730, A048733, A342698, A342700.

%K nonn,base,easy

%O 0,7

%A _Rémy Sigrist_, Mar 18 2021