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 A035513 Wythoff array read by antidiagonals. 133
 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-xy2 the determinant of any n X n contiguous subarray of A035513 (as a square array) is 0. - Gerald McGarvey, Sep 18 2004 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 = ABBA Wythoff sequence (Column 6) = A035340 = BBBA Wythoff sequence 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 All columns of the Wythoff array are compound Wythoff sequences. This follows from the main theorem in the 1972 paper by Carlitz,  Scoville and  Hoggatt. For an explicit expression see Theorem 10 in Kimberling's paper from 2008 in JIS. - Michel Dekking, Aug 31 2017 The Wythoff array can be viewed as an infinite graph over the set of nonnegative integers, built as follows: start with an empty graph; for all n = 0, 1, ..., create an edge between n and the sum of the degrees of all i < n. Finally, remove vertex 0. In the resulting graph, the connected components are chains and correspond to the rows of the Wythoff array. - Luc Rousseau, Sep 28 2017 REFERENCES J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996. C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..5151 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations, Fib. Quart. 10 (1972), 1-28. 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. John Conway, Alex Ryba, The extra Fibonacci series and the Empire State Building, Math. Intelligencer 38 (2016), no. 1, 41-48. 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. C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273. 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 FORMULA T(n, k) = Fib(k+1)*floor[n*tau]+Fib(k)*(n-1) where tau = (sqrt(5)+1)/2 = A001622 and Fib(n) = A000045(n). - Henry Bottomley, Dec 10 2001 T(n,-1) = n-1. T(n,0) = floor(n*tau). T(n,k) = T(n,k-1) + T(n,k-2) for k>=1. - R. J. Mathar, Sep 03 2016 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 (Python) from sympy import fibonacci as F, sqrt import math tau = (sqrt(5) + 1)/2 def T(n, k): return F(k + 1)*int(math.floor(n*tau)) + F(k)*(n - 1) for n in xrange(1, 11): print [T(k, n - k + 1) for k in xrange(1, n + 1)] # Indranil Ghosh, Apr 23 2017 CROSSREFS See comments above for more cross-references. Cf. A003622, A064274 (inverse), A083412 (transpose), A000201, A001950, A080164, A003603, A265650, A019586 (row that contains n). For two versions of the extended Wythoff array, see A287869, A287870. Sequence in context: A127008 A199535 A064274 * A191442 A191738 A218602 Adjacent sequences:  A035510 A035511 A035512 * A035514 A035515 A035516 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS Comments about the extended Wythoff array added by N. J. A. Sloane, Mar 07 2016 STATUS approved

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)