A080164 Wythoff difference array, D={d(i,j)}, by antidiagonals.

1, 2, 3, 5, 7, 4, 13, 18, 10, 6, 34, 47, 26, 15, 8, 89, 123, 68, 39, 20, 9, 233, 322, 178, 102, 52, 23, 11, 610, 843, 466, 267, 136, 60, 28, 12, 1597, 2207, 1220, 699, 356, 157, 73, 31, 14, 4181, 5778, 3194, 1830, 932, 411, 191, 81, 36, 16, 10946, 15127, 8362, 4791, 2440

Offset: 1

Author

Clark Kimberling, Feb 08 2003

Keywords

• nonn
• tabl

Comments

D is an interspersion formed by differences between Wythoff pairs in the Wythoff array W={w(i,j)}=A035513 (indexed so that i and j start at 1): d(i,j)=w(i,2j)-w(i,2j-1).

The difference between adjacent column terms is a Fibonacci number: d(i+1,j)-d(i,j) is F(2j) or F(2j+1).

Every term in column 1 of W is in column 1 of D; moreover, in row i of D, every term except the first is in row i of W.

Let W' be the array remaining when all the odd-numbered columns of W are removed from W. The rank array of W' (obtained by replacing each w'(i,j) by its rank when all the numbers w'(h,k) are arranged in increasing order) is D.

Let W" be the array remaining when all the even-numbered columns of W are removed from W; the rank array of W" is D.

Let D' be the array remaining when column 1 of D is removed; the rank array of D' is D.

Let E be the array {e(i,j)} given by e(i,j)=d(i,2j)-d(i,2j-1); the rank array of E is D.

D is the dispersion of the sequence u given by u(n)=n+floor(n*x), where x=(golden ratio); that is, D is the dispersion of the upper Wythoff sequence, A001950. For a discussion of dispersions, see A191426.

In column 1, F(2n) is in position F(2n-1) - Clark Kimberling, Jul 15 2016

Examples

			Northwest corner:
1   2   5   13   34   89
3   7   18  47   123  322
4   10  26  68   178  466
6   15  39  102  267  699
8   20  52  136  356  932
9   23  60  157  411  1076
		

References

• Clark Kimberling, The Wythoff difference array, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 153-158.

Links

• Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Visions, Games of No Chance 5 (2017) Vol. 70, See p. 65.
• Clark Kimberling, Interspersions
• Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
• Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A035513, A000201, A001950, A000045.

Programs

Mathematica

(* program generates the dispersion array T of the complement of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
x = 1 + GoldenRatio; f[n_] := Floor[n*x]
(* f(n) is complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A080164 as an array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A080164 as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011, added here by Clark Kimberling, Jun 03 2011 *)

Formula

d(i, j)=[i*tau]F(2j-1)+(i-1)F(2j-2), where F=A000045 (Fibonacci numbers). d(i, j)=[tau*d(i, j-1)]+d(i, j-1) for i>=2. d(i, j)=3d(i, j-1)-d(i, j-2) for i>=3.