A161674 Positions n such that A010060(n) + A010060(n+2) = 1.

0, 1, 4, 5, 6, 7, 8, 9, 12, 13, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 44, 45, 48, 49, 52, 53, 54, 55, 56, 57, 60, 61, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 104

Offset: 1

Author

Vladimir Shevelev, Jun 16 2009

Keywords

• nonn
• easy

Comments

Locates patterns of the form 0x1 or 1x0 in the Thue-Morse sequence.

Complement to A081706. Also: union of sequences {2*A121539(n)+k}, k=0 or 1, generalized in A161673.

Also union of sequences {A079523(n)-k}, k=0 or 1. For a generalization see A161890. - Vladimir Shevelev, Jul 05 2009

The asymptotic density of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
• Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.

Crossrefs

Cf. A161673, A161639, A161641, A161627, A161579, A161580, A161890, A121539, A131323, A036554, A010060, A079523, A081706.

Programs

Mathematica

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 6000, n++, If[tm[n] + tm[n + 2] == 1, Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
Flatten[Position[Partition[ThueMorse[Range[0,120]],3,1],?(#[[1]]+#[[3]] == 1&),1,Heads->False]]-1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 29 2019 *)

PARI

is(n)=hammingweight(n)%2!=hammingweight(n+2)%2 \\ Charles R Greathouse IV, Aug 20 2013

Extensions

Extended by R. J. Mathar, Aug 28 2009