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A128440
Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.
5
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 03 2007
STATUS
approved