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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A030574 a(n)=least k such that base 6 representation of n begins at s(k), where s=A030567.

Original entry on oeis.org

1, 2, 3, 4, 5, 17, 7, 1, 11, 13, 15, 29, 10, 21, 2, 25, 27, 41, 12, 24, 35, 3, 39, 53, 14, 26, 38, 49, 4, 5, 16, 28, 40, 52, 63, 173, 69, 17, 119, 137, 155, 68, 7, 8, 91, 94, 97, 71, 9, 20, 1, 143, 161, 74, 11, 32, 128, 146, 164, 77, 13, 44, 131
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030576 Length of n-th run of digit 1 in A030567.

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030577 Length of n-th run of digit 2 in A030567.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030578 Length of n-th run of digit 3 in A030567.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 5, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030579 Length of n-th run of digit 4 in A030567.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 5, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030582 a(n)=(# 1's)-(# 2's) in first n terms of A030567.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 3, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 6, 5, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10, 11
Offset: 1

Views

Author

Clark Kimberling

Keywords

Programs

  • Mathematica
    Accumulate[Which[#==1,1,#==2,-1,True,0]&/@Flatten[Table[ Reverse[ IntegerDigits[ n,6]],{n,0,50}]]] (* Harvey P. Dale, Sep 15 2019 *)

A030583 a(n)=(# 2's)-(# 3's) in first n terms of A030567.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 6, 5, 5, 4, 5, 4, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030584 (# 3's)-(# 4's) in first n terms of A030567.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 4, 5, 5, 6, 6, 5, 5, 4, 4, 3, 4, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Programs

  • Mathematica
    Accumulate[Which[#==3,1,#==4,-1,True,0]&/@Flatten[Table[Reverse[IntegerDigits[n,6]],{n,50}]]] (* Harvey P. Dale, Aug 07 2025 *)

A030308 Triangle T(n, k): Write n in base 2, reverse order of digits, to get the n-th row.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Comments

This is the quite common, so-called "bittest" function, see PARI code. - M. F. Hasler, Jul 21 2013
For a given number m and a digit position k the corresponding sequence index n can be calculated by n(m, k) = m*(1 + floor(log_2(m))) - 2^(1 + floor(log_2(m))) + k + 1. For example: counted from right to left, the second digit of m = 13 (binary 1101) is '0'. Hence the sequence index is n = n(13, 2) = 39. - Hieronymus Fischer, May 05 2007
A070939(n) is the length of n-th row; A000120(n) is the sum of n-th row; A030101(n) is the n-th row seen as binary number; A000035(n) = T(n, 0). - Reinhard Zumkeller, Jun 17 2012

Examples

			Triangle begins :
0
1
0, 1
1, 1
0, 0, 1
1, 0, 1
0, 1, 1
1, 1, 1
0, 0, 0, 1
1, 0, 0, 1 - _Philippe Deléham_, Oct 12 2011

Links

Crossrefs

Cf. A030190.
Cf. A030341, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-3 to base-10 analogs.

Programs

  • Haskell
    a030308 n k = a030308_tabf !! n !! k
a030308_row n = a030308_tabf !! n
a030308_tabf = iterate bSucc [0] where
   bSucc []       = [1]
   bSucc (0 : bs) = 1 : bs
   bSucc (1 : bs) = 0 : bSucc bs
-- Reinhard Zumkeller, Jun 17 2012
  • Maple
    A030308_row := n -> op(convert(n,base, 2)):
seq(A030308_row(n), n=0..23); # Peter Luschny, Nov 28 2017
  • Mathematica
    Flatten[Table[Reverse[IntegerDigits[n, 2]], {n, 0, 23}]] (* T. D. Noe, Oct 12 2011 *)
  • PARI
    A030308(n,k)=bittest(n,k) \\ Assuming that columns are numbered starting with k=0, as suggested by the formula from R. Zumkeller. - M. F. Hasler, Jul 21 2013
  • Python
    for n in range(20): print([int(z) for z in str(bin(n)[2:])[::-1]]) # Indranil Ghosh, Mar 31 2017
  • Sage
    A030308_row = lambda n: n.bits() if n > 0 else [0]
for n in (0..23): print(A030308_row(n)) # Peter Luschny, Nov 28 2017
  • Scala
    (0 to 31).map(Integer.toString(, 2).reverse).mkString.split("").map(Integer.parseInt()).toList // Alonso del Arte, Feb 10 2020

Formula

a(n) = floor(m/2^(k - 1)) mod 2, where m = max(j|A001855(j) < n) and k = n - A001855(m). - Hieronymus Fischer, May 05 2007, Sep 10 2007
T(n, k) = (n // 2^k) mod 2, for 0 <= k <= log[2](n) and n > 0; T(0, 0) = 0. ('//' denotes integer division). - Peter Luschny, Apr 20 2023

Extensions

Initial 0 and better name by Philippe Deléham, Oct 12 2011

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle begins :
0
1
2
0, 1
1, 1
2, 1
0, 2
1, 2
2, 2
0, 0, 1
1, 0, 1
2, 0, 1
0, 1, 1
1, 1, 1
2, 1, 1 ...

Links

Crossrefs

Cf. A081604 (row lengths), A053735 (row sums), A007089, A003137.
Cf. A030308, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Programs

  • Haskell
    a030341 n k = a030341_tabf !! n !! k
a030341_row n = a030341_tabf !! n
a030341_tabf = iterate succ [0] where
   succ []     = [1]
   succ (2:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Feb 21 2013
  • Maple
    A030341_row := n -> op(convert(n, base, 3)):
seq(A030341_row(n), n=0..32); # Peter Luschny, Nov 28 2017
  • Mathematica
    Flatten[Table[Reverse[IntegerDigits[n,3]],{n,0,40}]] (* Harvey P. Dale, Oct 20 2014 *)
  • PARI
    A030341(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\3^k%3 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Extensions

Initial 0 and better name by Philippe Deléham, Oct 20 2011
Previous Showing 11-20 of 28 results. Next

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