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%I #183 Jun 04 2024 15:36:22
%S 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,
%T 23,17,34,76,68,63,52,37,28,19,55,123,110,102,84,60,45,31,22,89,199,
%U 178,165,136,97,73,50,36,25,144,322,288,267,220,157,118,81,58,41,27,233,521
%N Wythoff array read by antidiagonals.
%C 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
%C Inverse of sequence A064274 considered as a permutation of the nonnegative integers. - _Howard A. Landman_, Sep 25 2001
%C 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
%C 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
%C From _Clark Kimberling_, Nov 14 2007: (Start)
%C Except for initial terms in some cases:
%C (Row 1) = A000045
%C (Row 2) = A000032
%C (Row 3) = A006355
%C (Row 4) = A022086
%C (Row 5) = A022087
%C (Row 6) = A000285
%C (Row 7) = A022095
%C (Row 8) = A013655 (sum of Fibonacci and Lucas numbers)
%C (Row 9) = A022112
%C (Row 10-19) = A022113, A022120, A022121, A022379, A022130, A022382, A022088, A022136, A022137, A022089
%C (Row 20-28) = A022388, A022096, A022090, A022389, A022097, A022091, A022390, A022098, A022092
%C (Column 1) = A003622 = AA Wythoff sequence
%C (Column 2) = A035336 = BA Wythoff sequence
%C (Column 3) = A035337 = ABA Wythoff sequence
%C (Column 4) = A035338 = BBA Wythoff sequence
%C (Column 5) = A035339 = ABBA Wythoff sequence
%C (Column 6) = A035340 = BBBA Wythoff sequence
%C Main diagonal = A020941. (End)
%C 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
%C 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
%C 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
%C 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
%C Suppose that h < k are consecutive terms in a row of the Wythoff array. If k is in an even numbered column, then h = floor(k/tau); otherwise, h = -1 + floor(k/tau). - _Clark Kimberling_, Mar 05 2020
%C From _Clark Kimberling_, May 26 2020: (Start)
%C For k > = 0, column k shows the numbers m having F(k+1) as least term in the Zeckendorf representation of m. For n >= 1, let r(n,k) be the number of terms in column k that are <= n. Then n/r(n,k) = n/(F(k+1)*tau + F(k)*(n-1)), by Bottomley's formula, so that the limiting ratio is 1/(F(k+1)*tau + F(k)). Summing over all k gives Sum_{k>=0} 1/(F(k+1)*tau + F(k)) = 1. Thus, in the limiting sense:
%C 38.19...% of the numbers m have least term 1;
%C 23.60...% have least term 2;
%C 14.58...% have least term 3;
%C 9.01...% have least term 5, etc. (End)
%C Named after the Dutch mathematician Willem Abraham Wythoff (1865-1939). - _Amiram Eldar_, Jun 11 2021
%D John H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.
%D Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
%H Alois P. Heinz, <a href="/A035513/b035513.txt">Table of n, a(n) for n = 1..5151</a>
%H Peter G. Anderson, <a href="https://www.fq.math.ca/Papers1/52-5/Anderson.pdf">More Properties of the Zeckendorf Array</a>, Fib. Quart. 52-5 (2014), 15-21.
%H L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-1/carlitz1.pdf">Fibonacci representations</a>, Fib. Quart., Vol. 10, No. 1 (1972), pp. 1-28.
%H Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/questions/183186/new-order-5-where-fibonacci-and-beatty-meet-at-wythoff">New Order #5: where Fibonacci and Beatty meet at Wythoff</a>.
%H J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, and N. J. A. Sloane, <a href="/A269725/a269725.txt">On Kimberling Sums and Para-Fibonacci Sequences</a>, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997.
%H John Conway and Alex Ryba, <a href="https://doi.org/10.1007/s00283-015-9582-5">The extra Fibonacci series and the Empire State Building</a>, Math. Intelligencer, Vol. 38, No. 1 (2016), pp. 41-48.
%H Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18. - _N. J. A. Sloane_, Jun 10 2012
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions</a>.
%H Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, JIS, Vol. 11 (2008), Article 08.3.3.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%H Clark Kimberling and Kenneth B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123, No. 2 (2016), pp. 267-273.
%H Stéphane Legendre, <a href="https://www.fq.math.ca/Papers1/53-2/Legendre10222014.pdf">Labeled Fibonacci Trees</a>, Fibonacci Quart. 53 (2015), no. 2, 152-167.
%H A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See pages 1-2.
%H Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, Vol. 41 (2013), pp. 175-192.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%H Sam Vandervelde, <a href="http://myslu.stlawu.edu/~svanderv/fibseqnorm.pdf">On the divisibility of Fibonacci sequences by primes of index two</a>, The Fibonacci Quarterly, Vol. 50, No. 3 (2012), pp. 207-216. See Figure 1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>.
%H Pedro Zanetti, <a href="/A035513/a035513.txt">C++ code snippet, for generating this sequence</a>.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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
%F 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
%e The Wythoff array begins:
%e 1 2 3 5 8 13 21 34 55 89 144 ...
%e 4 7 11 18 29 47 76 123 199 322 521 ...
%e 6 10 16 26 42 68 110 178 288 466 754 ...
%e 9 15 24 39 63 102 165 267 432 699 1131 ...
%e 12 20 32 52 84 136 220 356 576 932 1508 ...
%e 14 23 37 60 97 157 254 411 665 1076 1741 ...
%e 17 28 45 73 118 191 309 500 809 1309 2118 ...
%e 19 31 50 81 131 212 343 555 898 1453 2351 ...
%e 22 36 58 94 152 246 398 644 1042 1686 2728 ...
%e 25 41 66 107 173 280 453 733 1186 1919 3105 ...
%e 27 44 71 115 186 301 487 788 1275 2063 3338 ...
%e ...
%e 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:
%e 0 1 | 1 2 3 5 8 13 21 34 55 89 144 ...
%e 1 3 | 4 7 11 18 29 47 76 123 199 322 521 ...
%e 2 4 | 6 10 16 26 42 68 110 178 288 466 754 ...
%e 3 6 | 9 15 24 39 63 102 165 267 432 699 1131 ...
%e 4 8 | 12 20 32 52 84 136 220 356 576 932 1508 ...
%e 5 9 | 14 23 37 60 97 157 254 411 665 1076 1741 ...
%e 6 11 | 17 28 45 73 118 191 309 500 809 1309 2118 ...
%e 7 12 | 19 31 50 81 131 212 343 555 898 1453 2351 ...
%e 8 14 | 22 36 58 94 152 246 398 644 1042 1686 2728 ...
%e 9 16 | 25 41 66 107 173 280 453 733 1186 1919 3105 ...
%e 10 17 | 27 44 71 115 186 301 487 788 1275 2063 3338 ...
%e 11 19 | 30 49 79 ...
%e 12 21 | 33 54 87 ...
%e 13 22 | 35 57 92 ...
%e 14 24 | 38 62 ...
%e 15 25 | 40 65 ...
%e 16 27 | 43 70 ...
%e 17 29 | 46 75 ...
%e 18 30 | 48 78 ...
%e 19 32 | 51 83 ...
%e 20 33 | 53 86 ...
%e 21 35 | 56 91 ...
%e 22 37 | 59 96 ...
%e 23 38 | 61 99 ...
%e 24 40 | 64 ...
%e 25 42 | 67 ...
%e 26 43 | 69 ...
%e 27 45 | 72 ...
%e 28 46 | 74 ...
%e 29 48 | 77 ...
%e 30 50 | 80 ...
%e 31 51 | 82 ...
%e 32 53 | 85 ...
%e 33 55 | 88 ...
%e 34 56 | 90 ...
%e 35 58 | 93 ...
%e 36 59 | 95 ...
%e 37 61 | 98 ...
%e 38 63 | ...
%e ...
%e Each row of the extended Wythoff array also satisfies the Fibonacci recurrence, and may be extended to the left using this recurrence backwards.
%e From _Peter Munn_, Jun 11 2021: (Start)
%e The Wythoff array appears to have the following relationship to the traditional Fibonacci rabbit breeding story, modified for simplicity to be a story of asexual reproduction.
%e Give each rabbit a number, 0 for the initial rabbit.
%e When a new round of rabbits is born, allocate consecutive numbers according to 2 rules (the opposite of many cultural rules for inheritance precedence): (1) newly born child of Rabbit 0 gets the next available number; (2) the descendants of a younger child of any given rabbit precede the descendants of an older child of the same rabbit.
%e Row n of the Wythoff array lists the children of Rabbit n (so Rabbit 0's children have the Fibonacci numbers: 1, 2, 3, 5, ...). The generation tree below shows rabbits 0 to 20. It is modified so that each round of births appears on a row.
%e 0
%e :
%e ,-------------------------:
%e : :
%e ,---------------: 1
%e : : :
%e ,--------: 2 ,---------:
%e : : : : :
%e ,-----: 3 ,-----: ,-----: 4
%e : : : : : : : :
%e ,--: 5 ,--: ,---: 6 ,---: 7 ,---:
%e : : : : : : : : : : : : :
%e ,--: 8 ,--: ,--: 9 ,--: 10 ,--: ,--: 11 ,--: ,--: 12
%e : : : : : : : : : : : : : : : : : : : : :
%e : 13 : : 14 : 15 : : 16 : : 17 : 18 : : 19 : 20 :
%e The extended array's nontrivial extra column (A000201) gives the number that would have been allocated to the first child of Rabbit n, if Rabbit n (and only Rabbit n) had started breeding one round early.
%e (End)
%p 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
%p A035513 := proc(r, c)
%p option remember;
%p if c = 1 then
%p A003622(r) ;
%p else
%p A022342(1+procname(r, c-1)) ;
%p end if;
%p end proc:
%p seq(seq(A035513(r,d-r),r=1..d-1),d=2..15) ; # _R. J. Mathar_, Jan 25 2015
%t 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
%o (PARI) T(n,k)=(n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)
%o for(k=0,9,for(n=1,k, print1(T(n,k+1-n)", "))) \\ _Charles R Greathouse IV_, Mar 09 2016
%o (Python)
%o from sympy import fibonacci as F, sqrt
%o import math
%o tau = (sqrt(5) + 1)/2
%o def T(n, k): return F(k + 1)*int(math.floor(n*tau)) + F(k)*(n - 1)
%o for n in range(1, 11): print([T(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Apr 23 2017
%Y See comments above for more cross-references.
%Y Cf. A003622, A064274 (inverse), A083412 (transpose), A000201, A001950, A080164, A003603, A265650, A019586 (row that contains n).
%Y For two versions of the extended Wythoff array, see A287869, A287870.
%K nonn,tabl,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E Comments about the extended Wythoff array added by _N. J. A. Sloane_, Mar 07 2016