A003324 A nonrepetitive sequence. (Formerly M0443)

1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 4, 1, 2, 3, 4, 1, 4, 3, 4, 1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 3, 4, 1, 2, 3, 2, 1, 4, 3, 4, 1, 2, 3, 4, 1

Offset

1,2

Comments

Let b(0) be the sequence 1,2,3,4. Proceeding by induction, let b(n) be a sequence of length 2^(n+2). Quarter b(n) into four blocks, A,B,C,D each of length 2^n, so that b(n) = ABCD. Then b(n+1) = ABCDADCB. [After Dean paper.] - Sean A. Irvine, Apr 20 2015

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Richard A. Dean, A sequence without repeats on x, x^{-1}, y, y^{-1}, Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.

Françoise Dejean, Sur un Théorème de Thue, J. Combinatorial Theory, vol. 13 A, iss. 1 (1972) 90-99.

N. J. A. Sloane, P. Flor, L. F. Meyers, G. A. Hedlund. M. Gardner, Collection of documents and notes related to A1285, A3270, A3324

Jianing Song, Proof for my formula for A003324

Formula

a(n) = n mod 4 for odd n; for even n, write n = (2*k+1) * 2^e, then a(n) = 2 if k+e is odd, 4 if k+e is even. - Jianing Song, Apr 15 2021

Conjecture: a(2*n) = (A292077(n)+1)*2. Confirmed for first 1000 terms. - John Keith, Apr 18 2021 [This conjecture is correct. Write n = (2*k+1) * 2^e. If k+e is even, then we have A292077(n) = 0 and a(2n) = 2; if k+e is odd, then we have A292077(n) = 1 and a(2n) = 4. - Jianing Song, Nov 27 2021]

Mathematica

b[0] = Range[4];
b[n_] := b[n] = Module[{aa, bb, cc, dd}, {aa, bb, cc, dd} = Partition[b[n - 1], 2^(n-1)]; Join[aa, bb, cc, dd, aa, dd, cc, bb] // Flatten];
b[5] (* Jean-François Alcover, Sep 27 2017 *)
a[n_] := If[OddQ[n], Mod[n, 4], Module[{e = IntegerExponent[n, 2], k}, k = (n/2^e - 1)/2; If[OddQ[k + e], 2, 4]]];
Array[a, 100] (* Jean-François Alcover, Apr 19 2021, after Jianing Song *)

Prog

(PARI) a(n) = if(n%2, n%4, my(e=valuation(n, 2), k=bittest(n, e+1)); if((k+e)%2, 2, 4)) \\ Jianing Song, Apr 15 2021

Crossrefs

Positions of 1's, 2's, 3's and 4's: A016813, A343500, A004767, A343501.

Cf. A292077.

Sequence in context: A343251 A327464 A318308 * A110630 A343321 A238883

Adjacent sequences: A003321 A003322 A003323 * A003325 A003326 A003327

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved