cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A161890 Numbers such that A010060(n) = A010060(n+9).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 41, 45, 47, 48, 50, 51, 52, 54, 55, 57, 61, 63, 64, 66, 67, 68, 70, 71, 73, 77, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 103, 105, 109, 111, 112, 114, 115, 116, 118, 120
Offset: 0

Views

Author

Vladimir Shevelev, Jun 21 2009

Keywords

Comments

Or union of intersection of A161639 and {A079523(n)-8} and intersection of A161673 and {A121539(n)-8}. In general, for a>=1, consider equations A010060(x+a)+A010060(x)=1, A010060(x+a)=A010060(x). Denote via B_a (C_a) the sequence of nonnegative solutions of the first (second) equation. Then we have recursions: B_(a+1) is the union of transactions 1) C_a and {A121539(n)-a}, 2) B_a and {A079523(n)-a}; C_(a+1) is the union of transactions 1) C_a and {A079523(n)-a}, 2) B_a and {A121539(n)-a}.
Conjecture. In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009
This conjecture was actually proved in a later version of the Shevelev arxiv article cited below, and it can also easily be proved by the Walnut prover. - Jeffrey Shallit, Oct 12 2022

Links

Crossrefs

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, 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 <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
SequencePosition[ThueMorse[Range[0,150]],{x_,,,_,,,_,,,x_}][[All,1]]-1 (* Harvey P. Dale, Feb 06 2023 *)
  • PARI
    is(n)=hammingweight(n)%2==hammingweight(n+9)%2 \\ Charles R Greathouse IV, Aug 20 2013

Extensions

Terms a(35) onward added by G. C. Greubel, Jan 05 2018

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1
Offset: 0

Views

Author

Vladimir Shevelev, Jun 23 2009

Keywords

Comments

See comment to A161916. 3-a(n) is the number of equalities of kind A010060(n+k) = 1-A010060(n), k=1,2,3.

Crossrefs

Cf. A161916, A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060.

Programs

Extensions

Missing a(24)=1 inserted by Georg Fischer, Jun 21 2024

A162311 Numbers such that A010060(n) = A010060(n+7).

Original entry on oeis.org

1, 3, 4, 5, 7, 10, 14, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 42, 46, 49, 51, 52, 53, 55, 58, 62
Offset: 1

Views

Author

Vladimir Shevelev, Jul 01 2009

Keywords

Comments

Or union of intersection of A161673 and {A121539(n)-7} and intersection of A161639 and {A079523(n)-7}.
Conjecture: In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009

Links

Crossrefs

Cf. A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706, A161916, A161974.

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 <= 20000, n++, If[tm[n] == tm[n + 7], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
  • PARI
    is(n)=hammingweight(n)%2==hammingweight(n+7)%2 \\ Charles R Greathouse IV, Aug 20 2013

A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 48, 52, 54, 56, 60, 64, 68, 70, 72, 76, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 144, 148, 150, 152, 156, 158, 160, 164, 166, 168, 172, 176, 180
Offset: 1

Views

Author

Vladimir Shevelev, Jul 08 2009

Keywords

Comments

Numbers n for which A010060(n+1) = A010060(n+2) = 1-A010060(n) and A010060(n+3) = A010060(n).
Or intersection of A121539, A161674, and A161579.

Links

Crossrefs

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

Programs

  • Mathematica
    Select[Range[3500], MatchQ[IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[Length[{b}]]] &] - 1 (* G. C. Greubel, Jan 05 2018 *)
With[{nn=200},Sort[Join[SequencePosition[ThueMorse[Range[0,nn]],{1,0,0,1}],SequencePosition[ ThueMorse[Range[0,nn]],{0,1,1,0}]]][[;;,1]]]-1 (* Harvey P. Dale, Aug 20 2024 *)
  • PARI
    is(n)=my(v=vector(4,i,hammingweight(n+i-1))); v[1]==v[4] && v[1]!=v[2] && v[1]!=v[3] \\ Charles R Greathouse IV, Aug 20 2013

Formula

a(n) = A079523(n) - 1.

Extensions

More readable definition from R. J. Mathar, Sep 16 2009
Showing 1-4 of 4 results.

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