A128440 Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.

1, 3, 2, 4, 5, 4, 6, 7, 8, 6, 8, 10, 12, 13, 11, 9, 13, 16, 20, 22, 17, 11, 15, 21, 27, 33, 35, 29, 12, 18, 25, 34, 44, 53, 58, 46, 14, 20, 29, 41, 55, 71, 87, 93, 76, 16, 23, 33, 47, 66, 89, 116, 140, 152, 122, 17, 26, 38, 54, 77, 107, 145, 187, 228, 245, 199

Offset

1,2

Comments

Row 1 = Lower Wythoff sequence = A000201; Row 2 = Upper Wythoff sequence = A001950; Column 1 = A014217 (after first term); T(n,n) = A128440(n). Every positive integer occurs exactly once in the first two rows.

Conjecture: rows 2n-1 and 2n are disjoint for every positive integer n. - Clark Kimberling, Nov 11 2022

Stronger conjecture: for any positive integer n, if the numbers in rows 2n-1 and 2n are jointly arranged in increasing order, and each number is replaced by its position in the ordering, then the resulting two rows are identical to the first two rows. - Clark Kimberling, Nov 13 2022

Links

Michel Marcus, Table of n, a(n) for n = 1..5050 (Antidiagonals n=1..100 of array, flattened).

Formula

T(k,n) = k*F(n-1) + floor(k*t*F(n)), where F=A000045, the Fibonacci numbers.

Example

Corner:
   1    3    4    6    8    9   11   12
   2    5    7   10   13   15   18   20
   4    8   12   16   21   25   29   33
   6   13   20   27   34   41   47   54
  11   22   33   44   55   66   77   88
  17   35   53   71   89  107  125  143
  29   58   87  116  145  174  203  232
  46   93  140  187  234  281  328  375

Mathematica

r = (1 + Sqrt[5])/2; t[k_, n_] := Floor[n*r^k];
Grid[Table[t[k, n], {k, 1, 10}, {n, 1, 20}]]
(* Clark Kimberling, Nov 11 2022 *)

Prog

(PARI) T(n, k) = floor(k*quadgen(5)^n);
matrix(7, 7, n, k, T(n, k)) \\ Michel Marcus, Nov 14 2022

Crossrefs

Cf. A000045, A001622, A000201, A001950, A128439, A358359.

Sequence in context: A070221 A020814 A286298 * A063201 A173258 A039858

Adjacent sequences: A128437 A128438 A128439 * A128441 A128442 A128443

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2007

Status

approved