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A035513 Wythoff array read by antidiagonals. 111
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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. -Clark Kimberling, Jun 03 2011

REFERENCES

C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.

Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..5151

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quaterly, Vol. 50, February 2012. - N. J. A. Sloane, Jun 10 2012

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

FORMULA

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

EXAMPLE

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 ...

.......

MAPLE

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); # Alois P. Heinz, Aug 18 2008

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James W. Scheid (s1147798(AT)cedarville.edu)

STATUS

approved

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Last modified April 24 00:39 EDT 2014. Contains 240947 sequences.