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A363674 T(n,k) is the decimal equivalent of the n-bit inverted Gray code for k; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

%I A363674 #40 Jul 14 2023 11:51:49
%S A363674 0,1,0,3,2,0,1,7,6,4,5,1,0,2,3,15,14,12,13,9,8,10,11,3,2,0,1,5,4,6,7,
%T A363674 31,30,28,29,25,24,26,27,19,18,16,17,21,20,22,23,7,6,4,5,1,0,2,3,11,
%U A363674 10,8,9,13,12,14,15,63,62,60,61,57,56,58,59,51,50,48
%N A363674 T(n,k) is the decimal equivalent of the n-bit inverted Gray code for k; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.
%C A363674 Row n is a permutation of {0, 1, ..., A000225(n)}.
%H A363674 Alois P. Heinz, <a href="/A363674/b363674.txt">Rows n = 0..14, flattened</a>
%H A363674 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gray_code">Gray code</a>
%F A363674 T(n,k) = 2^n - 1 - A003188(k) = A000225(n) - A003188(k).
%F A363674 Sum_{k=0..2^n-1} (-1)^k * T(n,k) = A063524(n).
%F A363674 T(n,0) = T(n+1,2^(n+1)-1) = A000225(n).
%F A363674 T(n,A000975(n)) = 0.
%F A363674 T(n,A097072(n)) = 1 for n >= 1.
%F A363674 T(n,k) = T(n-1,k) + 2^(n-1) for n >= 1 and 0 <= k < 2^(n-1).
%F A363674 T(n,k) = T(n-1,2^n-1-k) for n >= 1 and 2^(n-1) <= k < 2^n.
%F A363674 A000120(T(n,n)) = A236840(n).
%e A363674 Triangle T(n,k) begins:
%e A363674    0;
%e A363674    1,  0;
%e A363674    3,  2,  0,  1;
%e A363674    7,  6,  4,  5, 1, 0,  2,  3;
%e A363674   15, 14, 12, 13, 9, 8, 10, 11, 3, 2, 0, 1, 5, 4, 6, 7;
%e A363674   ...
%e A363674 T(n,k) written in n-bit binary begins:
%e A363674     ();
%e A363674      1,    0;
%e A363674     11,   10,   00,   01;
%e A363674    111,  110,  100,  101,  001,  000,  010,  011;
%e A363674   1111, 1110, 1100, 1101, 1001, 1000, 1010, 1011, 0011, 0010, 0000, ...;
%e A363674   ...
%p A363674 T:= (n, k)-> Bits[Xor](2^n-1-k, iquo(k, 2)):
%p A363674 seq(seq(T(n, k), k=0..2^n-1), n=0..6);
%Y A363674 Columns k=0-2 give: A000225, A000918 (for n>=1), A028399 (for n>=2).
%Y A363674 Row sums give A006516.
%Y A363674 Cf. A000120, A000975, A003188, A063524, A097072, A236840, A329278, A331105, A362160.
%K A363674 nonn,tabf,look
%O A363674 0,4
%A A363674 _Alois P. Heinz_, Jun 14 2023