A374354 - OEIS

%I #13 Jul 09 2024 02:21:26

%S 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 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 5,16,17,20,21,2,4,18,20,2,5,18,21,8,16,8,9,16,17

%N 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 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 Rémy Sigrist, <a href="/A374354/b374354.txt">Table of n, a(n) for n = 0..8118</a> (rows for n = 0..1023 flattened)

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

%F T(n, 0) = 0 iff n is a fibbinary number.

%F T(n, k) + T(n, A277561(n)-1-k) = n.

%F T(n, 0) = A374355(n).

%F T(n, A277561(n)-1) = A374356(n).

%F Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).

%e Triangle T(n, k) begins:

%e   n   n-th row

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

%e    0  0

%e    1  0, 1

%e    2  0, 2

%e    3  1, 2

%e    4  0, 4

%e    5  0, 1, 4, 5

%e    6  2, 4

%e    7  2, 5

%e    8  0, 8

%e    9  0, 1, 8, 9

%e   10  0, 2, 8, 10

%e   11  1, 2, 9, 10

%e   12  4, 8

%e   13  4, 5, 8, 9

%e   14  4, 10

%e   15  5, 10

%e   16  0, 16

%o (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 See A295989 and A374361 for similar sequences.

%Y Cf. A003714, A037800, A277561, A374355, A374356.

%K nonn,base,tabf

%O 0,5

%A _Rémy Sigrist_, Jul 06 2024