A035506 Stolarsky array read by antidiagonals.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

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

Examples

			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
		

References

• C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• 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.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A035513 (Wythoff array), A035507 (inverse Stolarsky array), A191426.

Main diagonal gives A035489.

Programs

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 *)

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

Formula

T(1,k) = 2*T(0,k+1); T(3,k) = 3*T(0,k+2). - M. F. Hasler, Nov 05 2014

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