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A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

%I A374354 #13 Jul 09 2024 02:21:26
%S A374354 0,0,1,0,2,1,2,0,4,0,1,4,5,2,4,2,5,0,8,0,1,8,9,0,2,8,10,1,2,9,10,4,8,
%T A374354 4,5,8,9,4,10,5,10,0,16,0,1,16,17,0,2,16,18,1,2,17,18,0,4,16,20,0,1,4,
%U A374354 5,16,17,20,21,2,4,18,20,2,5,18,21,8,16,8,9,16,17
%N A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).
%C A374354 In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.
%H A374354 Rémy Sigrist, <a href="/A374354/b374354.txt">Table of n, a(n) for n = 0..8118</a> (rows for n = 0..1023 flattened)
%H A374354 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A374354 T(n, 0) = 0 iff n is a fibbinary number.
%F A374354 T(n, k) + T(n, A277561(n)-1-k) = n.
%F A374354 T(n, 0) = A374355(n).
%F A374354 T(n, A277561(n)-1) = A374356(n).
%F A374354 Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).
%e A374354 Triangle T(n, k) begins:
%e A374354   n   n-th row
%e A374354   --  -----------
%e A374354    0  0
%e A374354    1  0, 1
%e A374354    2  0, 2
%e A374354    3  1, 2
%e A374354    4  0, 4
%e A374354    5  0, 1, 4, 5
%e A374354    6  2, 4
%e A374354    7  2, 5
%e A374354    8  0, 8
%e A374354    9  0, 1, 8, 9
%e A374354   10  0, 2, 8, 10
%e A374354   11  1, 2, 9, 10
%e A374354   12  4, 8
%e A374354   13  4, 5, 8, 9
%e A374354   14  4, 10
%e A374354   15  5, 10
%e A374354   16  0, 16
%o A374354 (PARI) row(n) = { my (r = [0], e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], r = concat([v + y | v <- r], [v + x | v <- r]); break;););); return (r); }
%Y A374354 See A295989 and A374361 for similar sequences.
%Y A374354 Cf. A003714, A037800, A277561, A374355, A374356.
%K A374354 nonn,base,tabf
%O A374354 0,5
%A A374354 _Rémy Sigrist_, Jul 06 2024