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A336434 Square array read by descending antidiagonals T(n,k): In the binary expansion of n, reverse the order of the bits in the same position as the active bits in A057716(k). 1
2, 4, 1, 1, 2, 3, 4, 4, 6, 4, 8, 2, 5, 1, 6, 1, 2, 6, 2, 5, 5, 8, 8, 10, 1, 3, 3, 7, 1, 2, 9, 4, 5, 6, 7, 8, 8, 2, 10, 4, 12, 3, 7, 8, 10, 1, 2, 3, 4, 5, 6, 7, 8, 12, 9, 8, 8, 10, 8, 12, 12, 14, 8, 9, 10, 11, 16, 4, 9, 4, 9, 6, 13, 1, 12, 12, 14, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

T(n,k) is the swapping of the positions of the bits in n according to the active bits in K, where K = A057716(k). The bit in the same position as the first active bit in K switches positions with the bit in the same position as the last active bit in K, the bit in the same position as the second active bit in K switches with the one in the same as the second to last position, and so on until all have swapped (without repeating).

Any sequence, f, of the form "reverse the order of the a-th, b-th, ... and z-th bits in n" can be expressed as f(n) = T(n,k), where A057716(k) = 2^a + 2^b + ... 2^z. As a result, this operation combines 1 or more bit-swapping operations, which could be useful for bit-manipulation in computer programming.

LINKS

Davis Smith, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array

S. E. Anderson, Bit Twiddling Hacks.

FORMULA

T(n,k) = A003987(n, Sum_{m=1..floor(M/2)} A003987(A030308(n,A133457(K,m)), A030308(n,A133457(K,M - (m - 1))))* (2^A133457(K,m) + 2^A133457(K,M - (m - 1)))), where K = A057716(k) and M = A000120(A057716(k)).

When A057716(k) = 2^A070939(n) - 1, T(n,k) = A030101(n).

When A057716(k) = 2^(A070939(n) - 1) - 1, T(n,k) = A059893(n).

EXAMPLE

The binary expansion of 18 is 10010_2 and the active bits in the binary expansion of A057716(22) = 27 = 11011_2 are 0, 1, 3, and 4. So, to get T(18,22), we swap the 0th and 4th bits and then the 1st and 3rd bits, which gives us T(18,22) = 9.

Square array T(n,k) begins:

  \k   1   2   3   4   5   6   7   8   9  10 ...

  n\

   1|  2   4   1   4   8   1   8   1   8   1 ...

   2|  1   2   4   2   2   8   2   2   2   8 ...

   3|  3   6   5   6  10   9  10   3  10   9 ...

   4|  4   1   2   1   4   4   4   8   4   4 ...

   5|  6   5   3   5  12   5  12   9  12   5 ...

   6|  5   3   6   3   6  12   6  10   6  12 ...

   7|  7   7   7   7  14  13  14  11  14  13 ...

   8|  8   8   8   8   1   2   1   4   1   2 ...

   9| 10  12   9  12   9   3   9   5   9   3 ...

  10|  9  10  12  10   3  10   3   6   3  10 ...

PROG

(PARI) A336434(n, k)={my(K=k+#binary(k+#binary(k)), P=select(Z->bittest(K, Z), [0..#binary(K)-1]), Q1=P[1..floor(#P/2)], Q2=Vecrev(P)[1..floor(#P/2)], Sum=vecsum(apply(p->if(bittest(n, Q1[p])!=bittest(n, Q2[p]), bitor(shift(1, Q1[p]), shift(1, Q2[p]))), [1..floor(#P/2)]))); bitxor(n, Sum)}

CROSSREFS

Cf. A000120, A003987, A030101, A030308, A057716, A059893, A070939, A080412 (column k=1), A133457.

Sequence in context: A094593 A327814 A188348 * A007738 A186520 A158570

Adjacent sequences:  A336431 A336432 A336433 * A336435 A336436 A336437

KEYWORD

nonn,tabl,base

AUTHOR

Davis Smith, Jul 21 2020

STATUS

approved

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Last modified June 21 07:27 EDT 2021. Contains 345358 sequences. (Running on oeis4.)