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 1-3 of 3 results.

A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 4, 8, 3, 5, 3, 5, 7, 9, 11, 21, 5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55, 8, 12, 10, 18, 12, 16, 18, 34, 8, 12, 10, 18, 12, 16, 18, 34, 18, 26, 24, 44, 22, 30, 32, 60, 30, 46, 36, 64, 50, 66, 76, 144, 13, 19, 17, 31, 17, 23
Offset: 0

Views

Author

Philippe Deléham, Nov 06 2013

Keywords

Comments

The rule for constructing the tree is the following:
.....x
.....|
.....y
..../ \
..y+x..3y-x
and the tree begins like this:
.........0......
.........|......
.........1......
......./ \....
......1.....3....
...../ \.../ \...
....2...2.4...8..
and so on.
Column 1 : 0, 1, 1, 2, 3, 5, 8, ... = A000045 (Fibonacci numbers).
Column 2 : 3, 2, 5, 7, 12, 19, 31, ... = A013655.
Column 3 : 4, 3, 7, 10, 17, 27, 44, ... = A022120.
Column 4 : 8, 5, 13, 18, 31, 49, 80, ... = A022138.
Column 5 : 7, 5, 12, 17, 29, 46, 75, ... = A022137.
Column 6 : 9, 7, 16, 23, 39, 62, 101, ... = A190995.
Column 7 : 11, 7, 18, 25, 43, 68, 111, ... = A206419.
Column 8 : 21, 13, 34, 47, 81, 128, 209, ... = ?
Column 9 : 11, 8, 19, 27, 46, 73, 119, ... = A206420.
The lengths of the rows are 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... = A011782 .
The final numbers in the rows are 0, 1, 3, 8, 21, 55, 144, ... = A001906.
The middle numbers in the rows are 1, 2, 5, 13, 34, 89, ... = A001519 .
Row sums for n>=1: 1, 4, 16, 64, 256, 1024, ... = 4^(n-1).

Examples

			The successive rows are:
  0
  1
  1, 3
  2, 2, 4, 8
  3, 5, 3, 5, 7, 9, 11, 21
  5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55
  ...

Links

Crossrefs

Cf. A230872, A230873.
Cf. A231330, A231331, A231335.

Programs

  • Haskell
    data Dtree = Dtree Dtree (Integer, Integer) Dtree
a230871 n k = a230871_tabf !! n !! k
a230871_row n = a230871_tabf !! n
a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where
   rows (Dtree left (x, y) right) =
        [(x, y)] : zipWith (++) (rows left) (rows right)
   deleham (x, y) = Dtree
           (deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))
-- Reinhard Zumkeller, Nov 07 2013
  • Maple
    T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*
      T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))
    end:
seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # Alois P. Heinz, Nov 07 2013
  • Mathematica
    T[n_, k_] := T[n, k] = If[k==1 && n<2, n, Function[d, r = Quotient[k+1, 2]; (1+2d) T[n-1, r] + (1-2d) T[n-2, Quotient[r+1, 2]]][Mod[k+1, 2]]];
Table[T[n, k], {n, 0, 7}, {k, 1, Max[1, 2^(n-1)]}] // Flatten (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

Extensions

Incorrect formula removed by Michel Marcus, Sep 23 2023

A231330 Table of distinct terms in rows of triangle A230871, in natural order.

Original entry on oeis.org

0, 1, 1, 3, 2, 4, 8, 3, 5, 7, 9, 11, 21, 5, 7, 11, 13, 17, 19, 23, 25, 29, 55, 8, 10, 12, 16, 18, 22, 24, 26, 30, 32, 34, 36, 44, 46, 50, 60, 64, 66, 76, 144, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 61, 65, 67, 71, 73, 77, 79, 83, 89, 95
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 07 2013

Keywords

Comments

A230872 gives the union of all rows;
A231335(n) = number of Fibonacci numbers in row n.

Examples

			Initial rows:
.  0:  0;
.  1:  1;
.  2:  1,3;
.  3:  2,4,8                     from A230871(3,*) = [2,2,4,8];
.  4:  3,5,7,9,11,21             from A230871(4,*) = [3,5,3,5,7,9,11,21];
.  5:  5,7,11,13,17,19,23,25,29,55;
.  6:  8,10,12,16,18,22,24,26,30,32,34,36,44,46,50,60,64,66,76,144.

Links

Crossrefs

Cf. A231331 (row lengths), A000045 (left edge), A001906 (right edge).

Programs

  • Haskell
    import Data.List (sort, nub)
a231330 n k = a231330_tabf !! n !! k
a231330_row n = a231330_tabf !! n
a231330_tabf = map (sort . nub) a230871_tabf

A230873 Numbers that do not appear in A230871.

Original entry on oeis.org

6, 14, 15, 20, 28, 38, 40, 42, 48, 51, 52, 54, 72, 78, 84, 86, 88, 90, 96, 102, 103, 108, 110, 113, 114, 120, 124, 125, 126, 127, 132, 138, 150, 156, 162, 164, 168, 174, 180, 197, 198, 204, 210, 216, 220, 222, 224, 228, 236, 238, 245, 248, 250, 252, 258, 262, 270, 276, 279, 285, 286, 295, 298, 304, 306, 310, 315, 316
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2013

Keywords

Comments

Complement of A230872.

Links

Crossrefs

Cf. A230871, A230872.

Programs

  • Haskell
    a230873 n = a230873_list !! (n-1)
a230873_list = f [0..] a230872_list where
   f (u:us) vs'@(v:vs) = if u == v then f us vs else u : f us vs'
-- Reinhard Zumkeller, Nov 08 2013
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.