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A035513 Wythoff array read by antidiagonals. 126
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

If u and v are finite sets of numbers in a row of the Wythoff array such that (product of all the numbers in u) = (product of all the numbers in v), then u = v.  See A160009 (row 1 products), A274286 (row 2), A274287 (row 3), A274288 (row 4).  - Clark Kimberling, Jun 17 2016

REFERENCES

J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.

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

C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

LINKS

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

J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, 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.

C. Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008) 08.3.3

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

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Classic Sequences

Sam Vandervelde, On the divisibility of Fibonacci sequences by primes of index two, The Fibonacci Quarterly 50.3 (2012): 207-216. See Figure 1.

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 ...

.......

The extended Wythoff array has two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:

0     1  |   1    2    3    5    8   13   21   34   55   89  144   ...

1     3  |   4    7   11   18   29   47   76  123  199  322  521   ...

2     4  |   6   10   16   26   42   68  110  178  288  466  754   ...

3     6  |   9   15   24   39   63  102  165  267  432  699 1131   ...

4     8  |  12   20   32   52   84  136  220  356  576  932 1508   ...

5     9  |  14   23   37   60   97  157  254  411  665 1076 1741   ...

6    11  |  17   28   45   73  118  191  309  500  809 1309 2118   ...

7    12  |  19   31   50   81  131  212  343  555  898 1453 2351   ...

8    14  |  22   36   58   94  152  246  398  644 1042 1686 2728   ...

9    16  |  25   41   66  107  173  280  453  733 1186 1919 3105   ...

10   17  |  27   44   71  115  186  301  487  788 1275 2063 3338   ...

11   19  |  30   49   79   ...

12   21  |  33   54   87   ...

13   22  |  35   57   92   ...

14   24  |  38   62   ...

15   25  |  40   65   ...

16   27  |  43   70   ...

17   29  |  46   75   ...

18   30  |  48   78   ...

19   32  |  51   83   ...

20   33  |  53   86   ...

21   35  |  56   91   ...

22   37  |  59   96   ...

23   38  |  61   99   ...

24   40  |  64   ...

25   42  |  67   ...

26   43  |  69   ...

27   45  |  72   ...

28   46  |  74   ...

29   48  |  77   ...

30   50  |  80   ...

31   51  |  82   ...

32   53  |  85   ...

33   55  |  88   ...

34   56  |  90   ...

35   58  |  93   ...

36   59  |  95   ...

37   61  |  98   ...

38   63  |     ...

   ...

Each row of the extended Wythoff array also satisfies the Fibonacci recurrence, and may be extended to the left using this recurrence backwards.

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

A035513 := proc(r, c)

    option remember;

    if c = 1 then

        A003622(r) ;

    else

        A022342(1+procname(r, c-1)) ;

    end if;

end proc: # R. J. Mathar, Jan 25 2015

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

PROG

(PARI) T(n, k)=(n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)

for(k=0, 9, for(n=1, k, print1(T(n, k+1-n)", "))) \\ Charles R Greathouse IV, Mar 09 2016

CROSSREFS

See comments above for more cross-references.

Cf. A003622, A064274 (inverse), A083412, A000201, A001950, A080164, A003603, A265650.

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

Extended Wythoff array added by N. J. A. Sloane, Mar 07 2016

STATUS

approved

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Last modified August 27 12:58 EDT 2016. Contains 275906 sequences.