A035507 Inverse Stolarsky array read by antidiagonals.
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
Offset: 0
Examples
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, ...
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
Programs
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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
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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 + 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 *)
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
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
Comments