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.

Previous Showing 31-37 of 37 results.

A162549 Table of Dyck words.

Original entry on oeis.org

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

Views

Author

Franklin T. Adams-Watters, Jul 05 2009

Keywords

Crossrefs

Can be regarded as a table of paths, lengths A162550, or a table of all paths of length 2n, lengths A162551. Corresponding sums are A072643 and A001791.
Cf. A014486, A063171, A000108.

A244317 n occurs A014138(n) times.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 18 2014

Keywords

Comments

For n >= 1, a(n) = 1 + the least k such that A014143(k) >= n.
Useful when computing A244314.

Crossrefs

Cf. A014138, A014143, A244314, A072643, A081288, A244160.

Programs

  • Mathematica
    Join[{0},Flatten[Table[#[[2]],#[[1]]]&/@With[{nn=6},Thread[{Join[Accumulate[ CatalanNumber[ Range[ nn]]]],Range[nn]}]]]] (* Harvey P. Dale, Sep 06 2023 *)
  • Scheme
    (define (A244317 n) (if (zero? n) n (let loop ((k 0)) (if (>= (A014143 k) n) (+ 1 k) (loop (+ 1 k))))))

Formula

For all n >= 0, a(A014143(n)) = n+1 and a(1+A014143(n)) = n+2.

A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 22 2022

Keywords

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))

Crossrefs

Positions of first appearances are A014137.
Leaves of the ordered tree are counted by A057514, standard A358371.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
Positions of 2's are A155587, indices of A020988.
The standard ranking of the ordered tree is A358523.
Nodes of the ordered tree are counted by A358551, standard A358372.
For standard instead of binary encoding we have A358379.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263, A057122, A358373, A358505, A358524.

Programs

  • Mathematica
    binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]

A358551 Number of nodes in the ordered rooted tree with binary encoding A014486(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2022

Keywords

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))

Crossrefs

Run-lengths are A000108.
Binary encodings are listed by A014486.
Leaves of the ordered tree are counted by A057514, standard A358371.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
For standard instead of binary encoding we have A358372.
The standard ranking of the ordered tree is A358523.
Depth of the ordered tree is A358550, standard A358379.
Cf. A000081, A001263, A057122, A358373, A358505, A358524.

Programs

  • Mathematica
    binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Count[bint[k],_,{0,Infinity}],{k,Select[Range[0,10000],binbalQ]}]

Formula

a(n) = A072643(n) + 1.

A370291 Triangular number T(n) = A000217(n) occurs C(n) = A000108(n) times.

Original entry on oeis.org

0, 1, 3, 3, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21
Offset: 0

Views

Author

Paolo Xausa, Feb 14 2024

Keywords

Examples

			Written as a triangle:
   0;
   1;
   3,  3;
   6,  6,  6,  6,  6;
  10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10;
  ...

Links

Crossrefs

Cf. A000108, A000217, A072643.
Row sums of A370221 (for n >= 1).
Row sums as triangle give A002457(n-1) for n>=1.

Programs

  • Maple
    T:= n-> n*(n+1)/2$binomial(2*n,n)/(n+1):
seq(T(n), n=0..5);  # Alois P. Heinz, Feb 16 2024
  • Mathematica
    Flatten[Array[Table[PolygonalNumber[#], CatalanNumber[#]] &, 7, 0]]

Formula

a(n) = A000217(A072643(n)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1/2)^(n-1)/(2^n-1) = 0.86233289403022175171... . - Amiram Eldar, Feb 17 2024

A153249 Nonzero terms of table A153250 collected into one sequence row by row.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 14, 11, 12, 13, 15, 14, 15, 16, 19, 16, 17, 18, 20, 19, 20, 21, 22, 23, 24, 25, 28, 37, 25, 26, 27, 29, 38, 28, 29, 30, 33, 39, 30, 31, 32, 34, 40, 33, 34, 35, 36, 41, 37, 38, 39, 42, 51, 39, 40, 41, 43, 52, 42, 43, 44, 47, 53, 44, 45, 46
Offset: 0

Views

Author

Antti Karttunen, Dec 22 2008

Keywords

Comments

Row n (starting from row 0) contains A072643(n)+1 terms: 1; 2,3; 4,5,6; 6,7,8; 9,10,11,14; 11,12,13,15; ...

A162550 2n repeated C_n times, where C_n = A000108(n) is a Catalan number.

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 05 2009

Keywords

Comments

Lengths of Dyck paths (A162549).

Crossrefs

Cf. A162549, A072643, A000108.

Formula

a(n) = 2 * A072643(n).
Previous Showing 31-37 of 37 results.

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