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.

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

A230872 Numbers that appear in A230871.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 49, 50, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2013

Keywords

Comments

Union of rows of triangle A231330. - Reinhard Zumkeller, Nov 08 2013

Links

Crossrefs

Cf. A230871, A230873 (complement).

Programs

  • Haskell
    a230872 n = a230872_list !! (n-1)
a230872_list = f [] a231330_tabf where
   f ws (xs:xss) = us ++ f (merge vs xs) xss where
     (us,vs) = span (< head xs) ws
   merge us [] = us
   merge [] vs = vs
   merge us'@(u:us) vs'@(v:vs)
        | u < v = u : merge us vs'
        | u > v = v : merge us' vs
        | otherwise = u : merge us vs
-- Reinhard Zumkeller, Nov 08 2013

A231331 Number of distinct terms in row n of triangle A230871.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 74, 130, 258, 473, 1007, 1830, 3912, 7093, 15233, 27831, 60458, 109555, 239039, 433654, 946849, 1709524, 3746021, 6750928
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 07 2013

Keywords

Comments

Length of row n in triangle A231330.

Crossrefs

Cf. A231335.

Programs

  • Haskell
    a231331 = length . a231330_row
  • PARI
    vf(v) = #Set(v);
lista(nn) = my(va=[0], vb=[1]); print1(vf(va), ", "); print1(vf(vb), ", "); for (n=2, nn, v = vector(2^(n-1), k, j=(k+1)\2; i=(j+1)\2; y=vb[j]; x=va[i]; if (k%2, y+x, 3*y-x)); print1(vf(v), ", "); va = vb; vb = v;); \\ Michel Marcus, Sep 23 2023

Extensions

a(23)-a(25) from Michel Marcus, Sep 23 2023

A231335 Number of distinct Fibonacci numbers in rows of triangle A230871.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 4, 3, 3, 5, 4, 3, 4, 6, 4, 5, 4, 5, 6, 5, 4, 6, 7, 4, 5
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 07 2013

Keywords

Comments

a(n) = Sum_{k=1..A231331(n)} A010056(A231330(n,k));
a(n) > 1 for n > 1.

Examples

			a(0) = #{0} = 1;
a(1) = #{1} = 1;
a(2) = #{1, 3} = 2;
a(3) = #{2, 8} = 2;
a(4) = #{3, 5, 21} = 3;
a(5) = #{5, 13, 55} = 3;
a(6) = #{8, 34, 144} = 3;
a(7) = #{13, 55, 89, 377} = 4;
a(8) = #{21, 233, 987} = 3;
a(9) = #{34, 610, 2584} = 3;
a(10) = #{55, 89, 377, 1597, 6765} = 5;
a(11) = #{89, 377, 4181, 17711} = 4;
a(12) = #{144, 10946, 46368} = 3;
a(13) = #{233, 1597, 28657, 121393} = 4;
a(14) = #{377, 987, 1597, 6765, 75025, 317811} = 6;
a(15) = #{610, 10946, 196418, 832040} = 4;
a(16) = #{987, 4181, 6765, 514229, 2178309} = 5.

Crossrefs

Cf. A000045, A010056, A230871, A231330, A231331.

Programs

  • Haskell
    a231335 = length . filter ((== 1) . a010056) . a231330_row
  • PARI
    isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
vf(v) = #select(isfib, Set(v));
lista(nn) = my(va=[0], vb=[1]); print1(vf(va), ", "); print1(vf(vb), ", "); for (n=2, nn, v = vector(2^(n-1), k, j=(k+1)\2; i=(j+1)\2; y=vb[j]; x=va[i]; if (k%2, y+x, 3*y-x)); print1(vf(v), ", "); va = vb; vb = v;); \\ Michel Marcus, Sep 23 2023

Extensions

a(19)-a(25) from Michel Marcus, Sep 23 2023
Showing 1-4 of 4 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.