OFFSET
0,2
COMMENTS
The inverse Stolarky array is the dispersion of the sequence u given by u(n) = floor(n*x + x + n + 1 - x/2), where x=(golden ratio). For a discussion of dispersions, see A191426.
LINKS
C. Kimberling, Interspersions
C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society 117 (1993) 313-321.
N. J. A. Sloane, Classic Sequences
FORMULA
The term in row n and column k of the inverse Stolarsky array has the following expression: a(n, k) = F(2k-3) - 1 - c1(n)*F(2k-4) + c2(n)*F(2k-2), where F is the Fibonacci sequence; c1(n)=1 if n=1, [(n-1)*tau] if n>1 (first column of the Inverse Stolarsky array) and c2(n) = c1(n) + 1 + floor((2*c1(n)+1)*tau/2) (second column of the Inverse Stolarsky array). tau = (1+sqrt(5))/2 and [] denotes the nearest integer function. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004
Also, the following recurrence holds: a(n, k) = 3*a(n, k-1) - a(n, k-2) + 1 with a(n, 1)=c1(n) and a(n, 2)=c2(n). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004
EXAMPLE
Top left hand corner of array:
1, 4, 12, 33, 88, 232, ...
2, 7, 20, 54, 143, 376, ...
3, 9, 25, 67, 177, 465, ...
5, 14, 38, 101, 266, 698, ...
6, 17, 46, 122, 321, 842, ...
MAPLE
with(combinat, fibonacci): gold:=(1+sqrt(5))/2: c1:=n->piecewise(n<>1, round((n-1)*gold), 1): c2:=n->c1(n)+floor((2*c1(n)+1)*gold/2)+1: inv_stol:=(n, k)->fibonacci(2*k-3)-1-c1(n)*fibonacci(2*k-4)+c2(n)*fibonacci(2*k-2): seq(seq(inv_stol(n+1-k, k), k=1..n), n=1..11); inv_stol2:=(n, k)->(1+c0(n))*fibonacci(2*k-3)+(1+floor((2*c0(n)+1)*gold/2))*fibonacci(2*k-2)-1:seq(seq(inv_stol2(n+1-k, k), k=1..n), n=1..11); # C. Ronaldo, Dec 31 2004
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 + x + n + 1 - x/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]]; (* the array T *)
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* Inverse Stolarsky array, A035507 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004
Mathematica program, extended example, and comments from Clark Kimberling, Jun 03 2011
STATUS
approved