The Wythoff Array and The Para-Fibonacci Sequence

The Wythoff array A035513 is shown below, to the right of the broken line. It has many wonderful properties, some of which are listed after the table. It is also related to a large number of sequences in the On-Line Encyclopedia.

 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

Some properties of the Wythoff array.

(For sources see the "References" below.)

• Construction (1): the two columns to the left of the broken line consist respectively of the nonnegative integers n, and the lower Wythoff sequence A000201, whose nth term is [(n+1)tau], where tau=(1+sqrt(5))/2. The rows are then filled in by the Fibonacci rule that each term is the sum of the two previous terms. The entry n in the first column is the index of that row.

• Two definitions: The Zeckendorf expansion of n is obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains; for example 100 = 89 + 8 + 3 (see A035514, A035515, A035516, A035517).
  The Fibonacci successor to (or left shift of) n, Sn, say, is found by replacing each F_i in the Zeckendorf expansion by F_i+1; for example the successor to 100 is S100 = 144 + 13 + 5 = 162. See A022342.

• Construction (2): the two columns to the left of the broken line read n, 1+Sn; then after the broken line the sequence is
  m       Sm       SSm       SSSm       SSSSm       ...  ,

  where m = n + 1 + Sn.

• Construction (3): Let {S1, S2, S3, S4, ...} = {2,3,5,7,8,10,11,...} be the sequence of Fibonacci successors A022342. The first column of the array consists of the numbers not in that sequence: 1,4,6,9,12,... (A007067). The rest of each row is filled in by repeatedly applying S.

• Construction (4): The entry in row n and column k is
  [ (n+1) tau ] F_k+2 + n F_k+1 ,

  where {F₀, F₁, F₂, F₃, ...} = {0,1,1,2,3,5,...} are the Fibonacci numbers A000045.

• 1. The first row of the Wythoff array consists of the Fibonacci sequence 1,2,3,5,8,... A000045
  2. Every row satisfies the Fibonacci recurrence;
  3. The leading term in each row is the smallest number not found in any earlier row;
  4. Every positive integer appears exactly once in the array;
  5. The terms in any row or column are monotonically increasing;
  6. Every positive Fibonacci-type sequence (i.e. satisfying a(n)=a(n-1)+a(n-2) and eventually positive) appears as some row of the array;
  7. The terms in any two rows alternate.
  
  There are infinitely many arrays with properties 1-7, see [Kim95a].

• Another especially interesting array with properties 1-7 is the Stolarsky array: A035506,

   1   2   3    5    8   13   21   34   55   89
   4   6  10   16   26   42   68  110  178  288
   7  11  18   29   47   76  123  199  322  521
   9  15  24   39   63  102  165  267  432  699
  12  19  31   50   81  131  212  343  555  898
  14  23  37   60   97  157  254  411  665 1076
  17  28  45   73  118  191  309  500  809 1309
  20  32  52   84  136  220  356  576  932 1508
  22  36  58   94  152  246  398  644 1042 1686
  25  40  65  105  170  275  445  720 1165 1885
  

• The kth column of the Wythoff array consists of the numbers whose Zeckendorf expansion ends with F_k.

• The nth term of the vertical para-Fibonacci sequence
  0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, ...

  (A019586 or, for the original form, A003603) gives the index (or parameter) of the row of the Wythoff array that contains n.

  This sequence also has some nice properties.
  A. If you delete the first occurrence of each number, the sequence is unchanged. Thus if we delete the red numbers from

  0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, ... we get
  0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, ...

  again!

  B. Between any two consecutive 0's we see a permutation of the first few positive integers, and these nest, so the sequence can be rewritten as:

       0
       0
       0                1
       0         2      1
       0     3   2      1      4
       0   5 3   2    6 1    7 4
       0 8 5 3 9 2 10 6 1 11 7 4 12
  

• The nth term of the horizontal para-Fibonacci sequence
  1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, ...

  (A035612) gives the index (or parameter) of the column of the Wythoff array that contains n. This sequence also has a very nice property (see the entry).

References

[Con96] J. H. Conway, Unpublished notes, 1996.
[FrKi94] A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.
[Kim91] C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991, and Vol. 18, March 1992, p.82-83.
[Kim93] C. Kimberling, Orderings of the set of all positive Fibonacci sequences, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.
[Kim93a] C. Kimberling, Interspersions and dispersions, Proc. Amer. Math. Soc. 117 (1993) 313-321.
[Kim94] C. Kimberling, The First Column of an Interspersion, Fibonacci Quarterly 32 (1994) 301-314.
[Kim95] C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
[Kim95a] C. Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995) 129-138.
[Kim95b] C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
[Kim97] C. Kimberling, Fractal Sequences and Interspersions, Ars Combinatoria, vol 45 p 157 1997.
[Mor80] D. R. Morrison, A Stolarsky array of Wythoff pairs, in A Collection of Manuscripts Related to the Fibonacci Sequence, Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
[Sto76] K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull., 19 (1976), 472-482.
[Sto77] K. B. Stolarsky, A set of generalized Fibonacci sequences such that each natural number belongs to exactly one, Fib. Quart., 15 (1977), 224.

Other Links

Clark Kimberling, Fractal sequences
Clark Kimberling, Interspersions
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Associated Sequences

Successive columns of the Wythoff array A035513 give sequences A000201 (just before the broken line);
A007065, A035336, A035337, A035338, A035339, A035340.
Successive rows give the Fibonacci numbers A000045, the Lucas numbers A000204, the doubled Fibonacci numbers A013588, the trebled Fibonacci numbers A022086, A022087, A000285, A022095, etc.
The main diagonal is A020941.

 

 

Losanitsch's Triangle

An analogue of Pascal's triangle that deserves to be better known.

										1
									1		1
								1		1		1
							1		2		2		1
						1		2		4		2		1
					1		3		6		6		3		1
				1		3		9		10		9		3		1
			1		4		12		19		19		12		4		1
		1		4		16		28		38		28		16		4		1
1			5		20		44		66		66		44		20		5	q

The rule for producing these numbers is essentially the same as for Pascal's triangle: each term is the sum of the two numbers immediately above it, except that (numbering the rows by n=0,1,2,... and the entries in each row by k=0,1,2,...) if n is even and k is odd - the red entries! - we subtract C(n/2-1,(k-1)/2).

Formally,

a(n,k)=a(n - 1,k - 1)+a(n - 1,k) - C(n/2 - 1,(k - 1)/2), where the last term is present only if n even, k odd.

Reference: S. M. Losanitsch, Die Isomerie-Arten ... Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

The sequence formed by reading the triangle by rows is A034851, and the successive diagonals are A000012, A004526, A002620, A005993, A005994, A005995, A018210, A018211, A018212, A018213, A018214. The central columns yield A034872, A032123, A005654. The row sums form A005418. The difference between Pascal's triangle and the Losanitsch triangle gives the triangle shown in A034852.

The even-numbered diagonals are the partial sums of the previous diagonals. A generating function for the (2m)-th diagonal is

Sum C( m + 1, 2i ) x ²ⁱ , i = 0,1,2,...
-------------------------------------------
{( 1 - x ) ( 1 - x ² ) } ^m+1

and that for the (2m+1)st diagonal is obtained by dividing that by 1-x.

For example, the 5th diagonal 1,3,12,28,66,126,... has generating function

( 1 + 3 x ² )
---------------------------
{ ( 1 - x ) ( 1 - x ² ) } ³.

 

 

Posets.

How many partially ordered sets are there with n elements? (Sequence A001035.)
If the points are distinguishable, i.e. labeled, then for n = 1, 2, 3, ... points the numbers are:

1, 3, 19, 219, 4231, 130023, 6129859, ...

At present these numbers are known up through 18 points. Click to see the full entry.

Some related sequences are:

• A000112 (unlabeled posets)
• A000798 (labeled topologies)
• A001930 (unlabeled topologies)

A selection of references:

• K. K.-H. Butler, A Moore-Penrose inverse for Boolean relation matrices, pp. 18-28 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
• K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
• C. Chaunier and N. Lygeros, Progres dans l'enumeration des posets, C. R. Acad. Sci. Paris 314 serie I (1992) 691-694.
• C. Chaunier and N. Lygeros, The Number of Orders with Thirteen Elements, Order 9:3 (1992) 203-204.
• C. Chaunier and N. Lygeros, Le nombre de posets a isomorphie pres ayant 12 elements. Theoretical Computer Science, 123 (1994), 89-94.
• J. C. Culberson and G. J. E. Rawlins, New Results from an Algorithm for Counting Posets, Order 7 (90/91), no 4, pp. 361-374.
• M. Erne, The Number of Posets with More Points Than Incomparable Pairs, Disc Math 105 (1992) 49-60.
• M. Erne, On the cardinalities of finite topologies and the number of antichains in partially ordered sets, Discr. Math. 35 (1981) 119-133.
• M. Erne and K. Stege, Counting finite posets and topologies, Order, vol. 8, pp. 247-265, 1991.
• J. W. Evans, F. Harary and M. S. Lynn; On the computer enumeration of finite topologies; Comm. Assoc. Computing Mach. 10 (1967), 295--298.
• R. Fraisse and N. Lygeros, Petits posets : denombrement, representabilite par cercles et compenseurs. C. R. Acad. Sci. Paris, 313 (1991), 417-420.
• D. Kleitman & B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
• Y. Koda (ykoda@rst.fujixerox.co.jp), The numbers of finite lattices and finite topologies, Bull. Institute Combinatorics and its Applications, Jan. 1984.
• N. Lygeros, Calculs exhaustifs sur les posets d'au plus 7 elements. SINGULARITE, vol.2 n4 p.10-24, April 1991.
• N. Lygeros and P. Zimmermann, Calculation of a(14)
• P. Renteln, On the enumeration of finite topologies, J. Combin., Inform & System Sci., vol 19 pp 201-206 1994.
• P. Renteln, Geometrical approaches to the enumeration of finite posets ..., Nieuw Archiv Wisk., vol 14 pp 349-371 1996.
• V. I. Rodionov, MR 83k:05010 T(12) and T0(12) calculated (in Russian).

 

 

Hadamard's maximal determinant problem:

What is the largest determinant of any n x n matrix with entries that are 0 and 1 ? (Sequence A003432.)
Here is the sequence (for n = 1, 2, ...):

1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, 25515, 131072, 327680, 1114112

The next term is believed to be 3411968, although this has not been formally proved.

Click to see the full entry. Quite a lot is known about the above problem. See for example the survey article by J. Brenner in the Amer. Math. Monthly, June/July 1972, p. 626, and further comments in the issues of Dec. 1973, Dec. 1975 and Dec. 1977.
If n+1 is divisible by 4, and a Hadamard matrix of order n exists, then f(n) = (n+1)^{(n+1)/2}/2^n.
There are 4 equivalent versions of the problem: find the max determinant of a matrix with entries that are:
o 0 or 1, or
o in the range 0 <= x <= 1, or
o -1 or 1, or
o in the range -1 <= x <= 1.

For the most up-to-date information, see the web site The Hadamard Maximal Determinant Problem maintained by W. P. Orrick and B. Solomon. (Note that their indexing differs from that used in the OEIS.)

 

 

Bell numbers:

Expand exp(e^x - 1) in powers of x, SUM B_n xⁿ / n!. The coefficients B_n are the Bell numbers (A000110):

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, ...

Click to see the full entry.

 

 

Motzkin numbers:

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, ... (A001006).
Like the Catalan numbers, the Motzkin numbers have many interpretations. For example:

• the number of ways to join n points on a circle by nonintersecting chords
• the number of paths from (0,0) to (n,0) that do not go below the horizontal axis and are made up of steps (1,1) (i. e. NE), (1,-1) (i. e. SE) and (1,0) (i.e. E).
• the number of sequences (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0 = s(n).

A selection of references:

• T. Motzkin, Relations between hypersurface cross ratios... Bull. Amer. Math. Soc., 54, 352-360, 1948.
• R. Donaghey, Restricted plane tree representations of four Motzkin-Catalan equations, J. Combin. Theory Ser. B, 22, 114-121, 1977.
• R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory Ser. A, 23, 291-301, 1977.
• E. Barcucci, R. Pinzani, and R. Sprugnoli, The Motzkin family, PU. M. A. Ser. A, 2, No. 3-4, 249-279, 1991.
• A. Kuznetsov, I. Pak, and A. Postnikov, Trees associated with the Motzkin numbers, J. Combin. Theory Ser. A, 76, 145-147, 1996.
• F. Bergeron et al., Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 267.
• Richard Stanley's home page, under Enumerative Combinatorics, Vol II (to be published), has a list of manifestations of Motzkin numbers.

Formulas:

• G.f.: (1 - x - (1-2*x-3*x^2)^(1/2) ) / (2*x^2).
• G.f. satisfies A(x) = 1 + xA(x) + x^2 A(x)^2.
• Recurrence: a(n) = (-1/2) SUM (-3)^a C(1/2,a) C(1/2, b); a+b=n+2, a>=0, b>=0.
• In Maple: seriestolist(series((1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2),x,40));
• In Mathematica: a[0]=1;a[n_Integer]:=a[n]=a[n-1]+Sum[a[k]*a[n-2-k],{k,0,n-2}]; Array[ a[#]&, 30 ]

 

 

Perfect numbers:

Numbers that are equal to the sum of every (smaller) number that divides them (A000396).
For example 6 is perfect because it is divisible by 1, 2 and 3, and 1 + 2 + 3 = 6.
The sequence of perfect numbers begins:

6, 28, 496, 8128, 33550336, 8589869056, 137438691328,
2305843008139952128, 2658455991569831744654692615953842176, ...

Only some thirty or so perfect numbers are known. These are some of the largest numbers that have ever been computed. Click to see the full entry.

 

 

Aronson's sequence:

1, 4, 11, 16, 24, 29, 33, 35, 39, 45, 47, 51, 56, 58, 62, 64, ... (A005224):

whose definition is:

t is the first, fourth, eleventh, ... letter of this sentence

Click to see the full entry.

For a sequel, see that paper that Benoit Cloitre, Matthew Vandermast and I wrote: Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.

 

 

Chess games:

In the early 1990's my colleague Ken Thompson computed the number of possible chess games with n moves, for n up through 7, specially for the OEIS - see A006494.

There are now several versions of this sequence, depending on exactly what is being counted. The preferred version (now known for n <= 10) is A048987:

1, 20, 400, 8902, 197281, 4865609, 119060324, ...

For other versions see the entry for chess games in the Index to the OEIS.