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A035513
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Wythoff array read by antidiagonals.
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106
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1, 2, 4, 3, 7, 6, 5, 11, 10, 9, 8, 18, 16, 15, 12, 13, 29, 26, 24, 20, 14, 21, 47, 42, 39, 32, 23, 17, 34, 76, 68, 63, 52, 37, 28, 19, 55, 123, 110, 102, 84, 60, 45, 31, 22, 89, 199, 178, 165, 136, 97, 73, 50, 36, 25, 144, 322, 288, 267, 220, 157, 118, 81, 58, 41, 27, 233, 521
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OFFSET
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1,2
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COMMENTS
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T(0,0)=1, T(0,1)=2,...; y^2-x^2-xy<y if and only if there exist (i,j) with x=T(i,2j) and y=T(i,2j+1) - Claude Lenormand (claude.lenormand(AT)free.fr), Mar 17 2001
Inverse of sequence A064274 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001
The Wythoff array W consists of all the Wythoff pairs (x(n),y(n)), where x=A000201 and y=A001950, so that W contains every positive integer exactly once. The differences T(i,2j+1)-T(i,2j) form the Wythoff difference array, A080164, which also contains every positive integer exactly once. - Clark Kimberling, Feb 08 2003
For n>2 the determinant of any n X n contiguous subarray of A035513 (as a square array) is 0. - Gerald McGarvey, Sep 18 2004
Comments from Clark Kimberling, Nov 14 2007 (Start): Except for initial terms in some cases:
(Row 1) = A000045
(Row 2) = A000032
(Row 3) = A006355
(Row 4) = A022086
(Row 5) = A022087
(Row 6) = A000285
(Row 7) = A022095
(Row 8) = A013655 (sum of Fibonacci and Lucas numbers)
(Row 9) = A022112
(Row 10-19) = A022113, A022120, A022121, A022379, A022130, A022382, A022088, A022136, A022137, A022089
(Row 20-28) = A022388, A022096, A022090, A022389, A022097, A022091, A022390, A022098, A022092
(Column 1) = A003622 = AA Wythoff sequence
(Column 2) = A035336 = BA Wythoff sequence
(Column 3) = A035337 = ABA Wythoff sequence
(Column 4) = A035338 = BBA Wythoff sequence
(Column 5) = A035339
(Column 6) = A035340
Main diagonal = A020941 (End)
The Wythoff array is the dispersion of the sequence given by floor(n*x+x-1), where x=(golden ratio). See A191426 for a discussion of dispersions. [From Clark Kimberling, Jun 3 2011]
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REFERENCES
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LARRY ERICKSEN AND PETER G. ANDERSON, PATTERNS IN DIFFERENCES BETWEEN ROWS IN k-ZECKENDORF ARRAYS, http://www.cs.rit.edu/~pga/k-zeck.pdf, 2012. - From N. J. A. Sloane, Jun 10 2012
C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..5151
C. Kimberling, Interspersions
C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
Eric Weisstein's World of Mathematics, Wythoff Array
Index entries for sequences that are permutations of the natural numbers
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FORMULA
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T(n, k) = Fib(k+1)*floor[n*tau]+Fib(k)*(n-1) where tau = (sqrt(5)+1)/2 and Fib(n) = A000045(n). - Henry Bottomley, Dec 10 2001
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EXAMPLE
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The Wythoff array begins:
...1....2....3....5....8...13...21...34...55...89..144 ...
...4....7...11...18...29...47...76..123..199..322..521 ...
...6...10...16...26...42...68..110..178..288..466..754 ...
...9...15...24...39...63..102..165..267..432..699.1131 ...
..12...20...32...52...84..136..220..356..576..932.1508 ...
..14...23...37...60...97..157..254..411..665.1076.1741 ...
..17...28...45...73..118..191..309..500..809.1309.2118 ...
..19...31...50...81..131..212..343..555..898.1453.2351 ...
..22...36...58...94..152..246..398..644.1042.1686.2728 ...
..25...41...66..107..173..280..453..733.1186.1919.3105 ...
..27...44...71..115..186..301..487..788.1275.2063.3338 ...
.......
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MAPLE
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W:= proc(n, k) Digits:= 100; (Matrix ([n, floor((1+sqrt(5))/2* (n+1))]). Matrix([[0, 1], [1, 1]])^(k+1))[1, 2] end: seq (seq (W(n, d-n), n=0..d), d=0..10); [From Alois P. Heinz, Aug 18 2008]
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MATHEMATICA
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W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; Table[ W[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten
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CROSSREFS
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Cf. A003622. See also comments above. Cf. A064274 (inverse), A083412.
Sequence in context: A127008 A199535 A064274 * A191442 A191738 A218602
Adjacent sequences: A035510 A035511 A035512 * A035514 A035515 A035516
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James W. Scheid (s1147798(AT)cedarville.edu)
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STATUS
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approved
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