Search: a231335 -id:a231335
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A230871
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Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.
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+10
6
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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
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OFFSET
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0,4
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A231330
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Table of distinct terms in rows of triangle A230871, in natural order.
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+10
5
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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
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OFFSET
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0,4
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COMMENTS
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A230872 gives the union of all rows;
A231335(n) = number of Fibonacci numbers in row n.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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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
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listen;
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OFFSET
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0,3
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COMMENTS
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Length of row n in triangle A231330.
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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