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A035507 Inverse Stolarsky array read by antidiagonals. 7
1, 2, 4, 3, 7, 12, 5, 9, 20, 33, 6, 14, 25, 54, 88, 8, 17, 38, 67, 143, 232, 10, 22, 46, 101, 177, 376, 609, 11, 27, 59, 122, 266, 465, 986, 1596, 13, 30, 72, 156, 321, 698, 1219, 2583, 4180, 15, 35, 80, 190, 410, 842, 1829, 3193, 6764, 10945, 16, 41, 93, 211, 499 (list; table; graph; refs; listen; history; text; internal format)
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.

REFERENCES

C. Kimberling, "Interspersions and dispersions," Proceedings of the American Mathematical Society 117 (1993) 313-321.

LINKS

Table of n, a(n) for n=0..59.

C. Kimberling, Interspersions

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); (Ronaldo)

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

Cf. A035506 (Stolarsky array), A191426.

Sequence in context: A207625 A238953 A238964 * A138612 A246680 A294244

Adjacent sequences:  A035504 A035505 A035506 * A035508 A035509 A035510

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

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

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Last modified February 20 20:12 EST 2018. Contains 299385 sequences. (Running on oeis4.)