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.

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A127245 Row sums of a signed Thue-Morse related triangle.

Original entry on oeis.org

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

Views

Author

Paul Barry, Jan 10 2007

Keywords

Comments

0's occur at positions indexed by the odious numbers given by A091855.

Crossrefs

Row sums of A127244.
Cf. A000069, A010060, A091855, A127246.

Programs

  • Mathematica
    a[n_] := Mod[Sum[Product[ThueMorse[i], {i, k+1, n}], {k, 0, n}], 2]; Array[a, 100, 0] (* Amiram Eldar, Aug 04 2023 *)

Formula

a(n) = A127246(n) mod 2.
a(n) = Sum_{k=0..n} ((-1)^(n-k) * Product_{j=0..n-k-1} A010060(n-j)).

Extensions

More terms from Amiram Eldar, Aug 04 2023

A354384 Difference sequence of A356133.

Original entry on oeis.org

2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Crossrefs

Cf. A026430, A356133, A091855 (positions of 2), A036554 (positions of 3), A091855 (positions of 4).

Programs

  • Mathematica
    u = Accumulate[1 + ThueMorse /@ Range[0, 200]]  (* A026430 *)
v = Complement[Range[Max[u]], u];  (* A356133 *)
Differences[v] (* A354384 *)

Formula

a(n) = A007413(n) + 1.
a(n) = A036580(n) + 2.

A094177 Sequence from an aptitude test that I cannot work out!

Original entry on oeis.org

4, 3, 4, 9, 21, 51
Offset: 1

Views

Author

Dominique Butterworth (dominique_butterworth(AT)hotmail.com), May 06 2004

Keywords

Comments

Is this an erroneous version of A249453? - Arkadiusz Wesolowski, Oct 29 2014

Formula

For n>0, a(n) = 2^(n+1)-A091855(n) = 3+A128543(n). - M. F. Hasler, Nov 01 2014

A127253 Product of number triangles A127243 and A127248.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Barry, Jan 10 2007

Keywords

Comments

Rows containing -1 entries are indexed by twice the odious numbers given by A091855.

Examples

			Triangle begins:
  1;
  0, 1;
 -1, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, -1, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Crossrefs

Row sums are A127254.
Cf. A000069, A091855, A127243, A127248.

Programs

  • Mathematica
    T1[n_, k_] := SeriesCoefficient[(1 + ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127243 *)
T2[n_, k_] := SeriesCoefficient[(1 - ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127248 *)
T[n_, k_] := Sum[T2[n, j]*T1[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 04 2023 *)

A248416 Rectangular array by antidiagonals: for n >= 0, row n gives the positions in the Thue-Morse sequence A010059 at which the first 2^n terms occur.

Original entry on oeis.org

1, 4, 1, 6, 4, 1, 7, 7, 7, 1, 10, 11, 13, 13, 1, 11, 13, 21, 25, 25, 1, 13, 16, 25, 41, 49, 49, 1, 16, 19, 31, 49, 81, 97, 97, 1, 18, 21, 37, 61, 97, 161, 193, 193, 1, 19, 25, 41, 73, 121, 193, 321, 385, 385, 1, 21, 28, 49, 81, 145, 241, 385, 641, 769, 769, 1, 24, 31, 55, 97, 161, 289, 481, 769, 1281, 1537, 1537, 1
Offset: 1

Views

Author

Clark Kimberling, Oct 06 2014

Keywords

Comments

Each row contains contains its successor as a proper subsequence.
Note that this supposes that the Thue-Morse sequence A010059 has offset 1, whereas the true offset is 0. So really the entries should all be reduced by 1. - N. J. A. Sloane, Jul 01 2016
Apparently T(n,3) = A004119(n+1) for n>0. Apparently T(n,4) = A083575(n) for n>0. - R. J. Mathar, Nov 06 2018

Examples

			Northwest corner, n>=0, k>=1:
   1    4    6    7   10   11   13   16   18   19
   1    4    7   11   13   16   19   21   25   28
   1    7   13   21   25   31   37   41   49   55
   1   13   25   41   49   61   73   81   97  109
   1   25   49   81   97  121  145  161  193  217
   1   49   97  161  193  241  289  321  385  433
   1   97  193  321  385  481  577  641  769  865
The Thue-Morse sequence A010059 begins with 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, from which we see that the first 4 terms (=1,0,0,1) occur at positions 1, 7, 13, ..., as indicated for row n=2.

Crossrefs

Cf. A010059 (Thue-Morse), A026147 (row 0), A091855 (row 1?), A157971 (row 2?),
Column 1 is essentially A004119 (or A181565).

Programs

  • Maple
    A010060 := proc(n)
    local i;
    add(i, i=convert(n, base, 2)) mod 2 ;
end proc:
A010059 := proc(n)
    1-A010060(n) ;
end proc:
A248416Off0 := proc(n,k)
    option remember ;
    local strtN,binpat,src,thue ;
    if k = 1 then
        strtN := 0 ;
    else
        strtN := 1+procname(n,k-1) ;
    end if;
    binpat := [seq(A010059(i),i=0..n-1)] ;
    for src from strtN do
        thue := [seq(A010059(i),i=src..src+nops(binpat)-1)] ;
        if binpat=thue then
            return src ;
        end if;
    end do:
end proc:
A248416 := proc(n,k)
    1+A248416Off0(2^n,k) ;
end proc:
for d from 1 to 11 do
    for k from d to 1 by -1 do
        printf("%d,",A248416(d-k,k)) ;
end do: # R. J. Mathar, Nov 06 2018
  • Mathematica
    z = 3000; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 20]; Length[u]
t[p_, q_] := t[p, q] = Table[u[[k]], {k, p, q}];
r[n_] := Select[Range[z], t[#, # + 2^(n - 1)] == t[1, 1 + 2^(n - 1)] &]
TableForm[Table[r[n], {n, 0, 10}]]

Extensions

Definitions and examples clarified. - R. J. Mathar, Nov 06 2018
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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