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 A080164 Wythoff difference array, D={d(i,j)}, by antidiagonals. 6
 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; text; internal format)
 OFFSET 1,2 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 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, Urban Larsson, Wythoff Visions, Games of No Chance 5 (2017) Vol. 70, See p. 65. C. Kimberling, Interspersions Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4. 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. 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 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 *) CROSSREFS Cf. A035513, A000201, A001950, A000045. Sequence in context: A191723 A292874 A302847 * A182949 A245711 A246258 Adjacent sequences:  A080161 A080162 A080163 * A080165 A080166 A080167 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 08 2003 STATUS approved

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Last modified October 15 18:22 EDT 2019. Contains 328037 sequences. (Running on oeis4.)