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A035506
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Stolarsky array read by antidiagonals.
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50
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1, 2, 4, 3, 6, 7, 5, 10, 11, 9, 8, 16, 18, 15, 12, 13, 26, 29, 24, 19, 14, 21, 42, 47, 39, 31, 23, 17, 34, 68, 76, 63, 50, 37, 28, 20, 55, 110, 123, 102, 81, 60, 45, 32, 22, 89, 178, 199, 165, 131, 97, 73, 52, 36, 25, 144, 288, 322, 267, 212, 157, 118, 84, 58, 40, 27, 233, 466, 521, 432, 343, 254, 191, 136, 94, 65, 44, 30
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Inverse of sequence A064357 considered as a permutation of the positive integers. - Howard A. Landman (howard(AT)polyamory.org), Sep 25 2001
GP-PARI program gives general solution for the Stolarsky array in square array form by row,column. Increase the default precision, if computing large values in the array. - Randall L. Rathbun (randallr(AT)abac.com), Jan 25 2002
The Stolarsky array is the dispersion of the sequence s given by s(n)=(integer nearest n*x), where x=(golden ratio). For a discussion of dispersions, see A191426.
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REFERENCES
| C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
C. Kimberling, "Interspersions and dispersions," Proceedings of the American Mathematical Society 117 (1993) 313-321.
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..10010
C. Kimberling, Interspersions
N. J. A. Sloane, Classic Sequences
Eric Weisstein's World of Mathematics, Stolarsky arrays
Index entries for sequences that are permutations of the natural numbers
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EXAMPLE
| Top left corner of array is:
1....2....3....5....13...21...34...55....89
4....6....10...16...26...42...68...110...178
7....11...18...29...47...76...123..119...322
9....15...24...39...63...102..165..267...432
12...19...31...50...81...131..212..343...555
14...23...37...60...97...157..254..411...665
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MAPLE
| A:= proc (n, k) local t, a, b; t:= (1+sqrt(5))/2; a:= floor (n*(1+t)-t/2); b:= round (a*t); (Matrix([[b, a]]). Matrix([[1, 1], [1, 0]])^(k-1))[1, 2] end: seq (seq (A (n, d-n), n=1..d-1), d=1..11); # Alois P. Heinz, Aug 17 2008
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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 = GoldenRatio; f[n_] := Floor[n*x + 1/2]
(* 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}]]
(* t=Stolarsky array, A035506 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* Stolarsky array as a sequence *)
(* Program by Peter Moses, June 1, 2011 *)
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PROG
| (PARI) {Stolarsky(r, c)= tau=(1+sqrt(5))/2; a=floor(r*(1+tau)-tau/2); b=round(a*tau); if(c==1, a, if(c==2, b, for(i=1, c-2, d=a+b; a=b; b=d; ); d))}
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CROSSREFS
| Cf. A035513 (Wythoff array),
A035507 (inverse Stolarksy array),
A191426.
Sequence in context: A194030 A083044 A126714 * A006016 A054239 A048680
Adjacent sequences: A035503 A035504 A035505 * A035507 A035508 A035509
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000.
Extended (terms, Mathematica, example) by Clark Kimberling, Jun 3 2011.
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