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%I #10 Apr 07 2023 02:35:29
%S 3,8,11,21,29,16,55,76,42,24,144,199,110,63,32,377,521,288,165,84,37,
%T 987,1364,754,432,220,97,45,2584,3571,1974,1131,576,254,118,50,6765,
%U 9349,5168,2961,1508,665,309,131,58,17711,24476,13530,7752,3948,1741,809
%N (1,2)-block array, B(1,2), of the Wythoff array (A035513), read by descending antidiagonals.
%C 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.
%F 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).
%F 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.
%e Corner of B(1,2):
%e 3 8 21 55 144 377 987 ...
%e 11 29 76 199 521 1364 3571 ...
%e 16 42 110 288 754 1974 5168 ...
%e 24 63 165 432 1131 2961 7752 ...
%e 32 84 220 576 1508 3948 10336 ...
%e ...
%e (row 1 of A035513) = (1,2,3,5,8,13,21,34,...), so (row 1 of B(1,2)) = (3,8,21,55,...);
%e (row 2 of A000027) = (4,7,11,18,29,47,76,123,...), so (row 2 of B(1,2)) = (11,29,76,199,...).
%t f[n_] := Fibonacci[n]; r = GoldenRatio;
%t zz = 10; z = 13;
%t w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
%t t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
%t Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A361992 sequence *)
%t TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361992 array *)
%Y Cf. A000045, A001622, A035513, A080164, A361974, A361993 (array B(2,1)), A361994 (array B(2,2)).
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Apr 04 2023