A329303 -id:A329303 - cp's OEIS frontend

A330091 Inverse permutation to A329303.

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 13, 14, 15, 16, 23, 22, 19, 18, 21, 20, 17, 24, 27, 26, 25, 28, 29, 30, 31, 32, 47, 46, 39, 38, 45, 44, 35, 34, 41, 42, 37, 36, 43, 40, 33, 48, 55, 54, 51, 50, 53, 52, 49, 56, 59, 58, 57, 60, 61, 62, 63, 64, 95, 94, 79

Offset: 0

Rémy Sigrist, Dec 01 2019

If the run lengths in binary expansion of n are (r(1), ..., r(w)), then the run lengths in binary expansion of a(n) are (r(1), r(w), r(2), r(w-1), ...); this corresponds to a "milk shuffle".

			A329303(43) = 45, hence a(45) = 43.

Cf. A329303.

torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2),2)); rr = concat(r, rr); n\=2^r); rr }
unshuffle(v) = { my (w=vector(#v), o=0, e=#v+1); for (k=1, #v, w[k]=v[if (k%2, o++, e--)]); w }
fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
a(n) = fromrl(unshuffle(torl(n)))

A330081 If the binary expansion of n is (b(1), ..., b(w)), then the binary expansion of a(n) is (b(1), b(3), b(5), ..., b(6), b(4), b(2)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 10, 12, 14, 9, 11, 13, 15, 16, 20, 18, 22, 24, 28, 26, 30, 17, 21, 19, 23, 25, 29, 27, 31, 32, 36, 40, 44, 34, 38, 42, 46, 48, 52, 56, 60, 50, 54, 58, 62, 33, 37, 41, 45, 35, 39, 43, 47, 49, 53, 57, 61, 51, 55, 59, 63, 64, 72, 68, 76

Offset: 0

Rémy Sigrist, Dec 01 2019

This sequence is a permutation of the nonnegative integers that preserves the binary length as well as the Hamming weight. See A330090 for the inverse.

			For n = 1234:
- the binary expansion of 1234 is "10011010010",
- odd-indexed bits are "101100",
- even-indexed bits are "01001", and in reverse order "10010",
- hence the binary expansion of a(1234) is "10110010010",
- so a(1234) = 1426.

See A329303 for a similar sequence.

Cf. A003558, A194959, A330090 (inverse).

shuffle(v) = { my (w=vector(#v), o=0, e=#v+1); for (k=1, #v, w[if (k%2, o++, e--)]=v[k]); w }
a(n) = fromdigits(shuffle(binary(n)), 2)

If n has w binary digits, then a^A003558(w-1)(n) = n (where a^k denotes the k-th iterate of the sequence).

cp's OEIS Frontend

A330091 Inverse permutation to A329303.

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