A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

5, 9, 15, 14, 25, 26, 23, 40, 43, 36, 37, 65, 69, 59, 47, 60, 105, 112, 95, 77, 57, 97, 170, 181, 154, 124, 93, 68, 157, 275, 293, 249, 201, 150, 111, 78, 254, 445, 474, 403, 325, 243, 179, 127, 89, 411, 720, 767, 652, 526, 393, 290, 205, 145, 99, 665, 1165

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(2,1) is a row-splitting array. The rows of B(2,1) are linearly recurrent with signature (1,1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,1) is A361995.

Links

Table of n, a(n) for n=1..57.

Formula

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

b(i,j) = F(j+1) ([2 i r] + [(2 i - 1) r]) + (4 i - 3) F(j), where F = A000045, the Fibonacci numbers, and r = (1+sqrt(5))/2, the golden ratio, A001622, and [ ] = floor.

Example

Corner of B(2,1):
   5    9   14   23   37   60   97  157 ...
  15   25   40   65  105  170  275  445 ...
  26   43   69  112  181  293  474  767 ...
  36   59   95  154  249  403  652 1055 ...
  47   77  124  202  325  526  851 1377 ...
  ...
(column 1 of A035513) = (1,4,6,9,12,14,17,19,...), so (column 1 of B(2,1)) = (5,15,26,36,...);
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so (column 2 of B(2,1)) = (9,25,43,59,...).

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[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A361993 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361993 array *)

Crossrefs

Cf. A000045, A001622, A035513, A080164, A361975, A361992 (array B(1,2)), A361994 (array B(2,2)).

Sequence in context: A331556 A188358 A228218 * A314993 A314994 A106503

Adjacent sequences: A361990 A361991 A361992 * A361994 A361995 A361996

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 04 2023

Status

approved