A361992 (1,2)-block array, B(1,2), of the Wythoff array (A035513), read by descending antidiagonals.

3, 8, 11, 21, 29, 16, 55, 76, 42, 24, 144, 199, 110, 63, 32, 377, 521, 288, 165, 84, 37, 987, 1364, 754, 432, 220, 97, 45, 2584, 3571, 1974, 1131, 576, 254, 118, 50, 6765, 9349, 5168, 2961, 1508, 665, 309, 131, 58, 17711, 24476, 13530, 7752, 3948, 1741, 809

Offset

1,1

Comments

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(1,2) is a row-splitting array. The rows of B(1,2) are linearly recurrent with signature (3,-1). The order array (as defined in A333029) of B(1,2) is the Wythoff difference array, A080164.

Links

Table of n, a(n) for n=1..52.

Formula

B(1,2) = (b(i,j)), where b(i,j) = w(i, 2j-1) + w(i, 2j) for i >= 1, j >= 1, where (w(i,j)) is the Wythoff array (A035513).

b(i,j) = w(i,2j+1) = F(2 k + 2)*floor(h r) + (h - 1)F(2 k + 1), where F = A000045, the Fibonacci numbers, and r = (1+sqrt(5))/2, the golden ratio, A001622.

Example

Corner of B(1,2):
   3     8    21    55   144   377   987 ...
  11    29    76   199   521  1364  3571 ...
  16    42   110   288   754  1974  5168 ...
  24    63   165   432  1131  2961  7752 ...
  32    84   220   576  1508  3948 10336 ...
  ...
(row 1 of A035513) = (1,2,3,5,8,13,21,34,...), so (row 1 of B(1,2)) = (3,8,21,55,...);
(row 2 of A000027) = (4,7,11,18,29,47,76,123,...), so (row 2 of B(1,2)) = (11,29,76,199,...).

Mathematica

f[n_] := Fibonacci[n]; r = GoldenRatio;
zz = 10; z = 13;
w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A361992 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361992 array *)

Crossrefs

Cf. A000045, A001622, A035513, A080164, A361974, A361993 (array B(2,1)), A361994 (array B(2,2)).

Sequence in context: A364086 A171672 A341262 * A070073 A334539 A058565

Adjacent sequences: A361989 A361990 A361991 * A361993 A361994 A361995

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 04 2023

Status

approved