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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.
4
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, 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, 10, 8, 9, 13, 12, 14, 15, 63, 62, 60, 61, 57, 56, 58, 59, 51, 50, 48
OFFSET
0,4
COMMENTS
Row n is a permutation of {0, 1, ..., A000225(n)}.
LINKS
Wikipedia, Gray code
FORMULA
T(n,k) = 2^n - 1 - A003188(k) = A000225(n) - A003188(k).
Sum_{k=0..2^n-1} (-1)^k * T(n,k) = A063524(n).
T(n,0) = T(n+1,2^(n+1)-1) = A000225(n).
T(n,A000975(n)) = 0.
T(n,A097072(n)) = 1 for n >= 1.
T(n,k) = T(n-1,k) + 2^(n-1) for n >= 1 and 0 <= k < 2^(n-1).
T(n,k) = T(n-1,2^n-1-k) for n >= 1 and 2^(n-1) <= k < 2^n.
A000120(T(n,n)) = A236840(n).
EXAMPLE
Triangle T(n,k) begins:
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(n,k) written in n-bit binary begins:
();
1, 0;
11, 10, 00, 01;
111, 110, 100, 101, 001, 000, 010, 011;
1111, 1110, 1100, 1101, 1001, 1000, 1010, 1011, 0011, 0010, 0000, ...;
...
MAPLE
T:= (n, k)-> Bits[Xor](2^n-1-k, iquo(k, 2)):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);
CROSSREFS
Columns k=0-2 give: A000225, A000918 (for n>=1), A028399 (for n>=2).
Row sums give A006516.
Sequence in context: A345066 A253669 A154477 * A322324 A142071 A350448
KEYWORD
nonn,tabf,look
AUTHOR
Alois P. Heinz, Jun 14 2023
STATUS
approved