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A035506 Stolarsky array read by antidiagonals. 53
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; text; internal format)
OFFSET
0,2
COMMENTS
Inverse of sequence A064357 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
The PARI/GP script gives a general solution for the Stolarsky array in square array form by row, column. Increase the default precision to compute large values in the array. - Randall L Rathbun, 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.
See A098861 for the row in which is a given number. - M. F. Hasler, Nov 05 2014
Named after the American mathematician Kenneth Barry Stolarsky. - Amiram Eldar, Jun 11 2021
REFERENCES
C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
LINKS
Clark Kimberling, Interspersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, Vol. 117 (1993), pp. 313-321.
David R. Morrison, A Stolarsky array of Wythoff pairs, A collection of manuscripts related to the Fibonacci sequence, Santa Clara, CA: Fibonacci Association, 1980, pp. 134-136.
N. J. A. Sloane, Classic Sequences.
Eric Weisstein's World of Mathematics, Stolarsky arrays.
FORMULA
T(1,k) = 2*T(0,k+1); T(3,k) = 3*T(0,k+2). - M. F. Hasler, Nov 05 2014
EXAMPLE
Top left corner of the array is:
1 2 3 5 8 13 21 34 55
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
MAPLE
A:= proc(n, k) local t, a, b; t:= (1+sqrt(5))/2; a:= floor(n*(t+1)+1 +t/2); b:= round(a*t); (Matrix([[b, a]]). Matrix([[1, 1], [1, 0]])^k) [1, 2] end: seq(seq(A (n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Aug 17 2008
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 J. C. Moses, Jun 01 2011 *)
(* Second program: *)
A[n_, k_] := Module[{t, a, b}, t = (1+Sqrt[5])/2; a = Floor[n*(t+1)+1+t/2]; b = Round[a*t]; ({b, a}.MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 22 2023, after Alois P. Heinz *)
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))} \\ Randall L Rathbun, Jan 25 2002
CROSSREFS
Cf. A035513 (Wythoff array), A035507 (inverse Stolarsky array), A191426.
Main diagonal gives A035489.
Sequence in context: A083044 A361995 A126714 * A246368 A316963 A320672
KEYWORD
nonn,tabl,easy,nice
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000
Extended (terms, Mathematica, example) by Clark Kimberling, Jun 03 2011
Example corrected by M. F. Hasler, Nov 05 2014
STATUS
approved

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Last modified February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)