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A080164
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Wythoff difference array, D={d(i,j)}, by antidiagonals.
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5
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 and 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.
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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.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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LINKS
| C. Kimberling, Interspersions
Index entries for sequences that are permutations of the natural numbers
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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.
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EXAMPLE
| 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
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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 Moses, Jun 1 2011, added here by Clark Kimberling, Jun 3 2011 *)
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CROSSREFS
| Cf. A035513, A000201, A001950, A000045.
Sequence in context: A103866 A191439 A191723 * A182949 A126048 A142349
Adjacent sequences: A080161 A080162 A080163 * A080165 A080166 A080167
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Feb 08 2003
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