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 A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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, 13, ... = 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). LINKS Reinhard Zumkeller, Rows n = 0..13 of triangle, flattened FORMULA T(n,k) = T(n-1,k) + T(n-2,k) for k < 2^(n-2), n > 1. - Philippe Deléham, Nov 07 2013 EXAMPLE 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 ... 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 *) PROG (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 CROSSREFS Cf. A230872, A230873. Cf. A231330, A231331, A231335. Sequence in context: A325596 A254876 A249159 * A111241 A247501 A192183 Adjacent sequences:  A230868 A230869 A230870 * A230872 A230873 A230874 KEYWORD nonn,tabf AUTHOR Philippe Deléham, Nov 06 2013 STATUS approved

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Last modified June 13 17:44 EDT 2021. Contains 345008 sequences. (Running on oeis4.)