Revision History for A361993
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#11 by Michael De Vlieger at Sun Jun 04 23:50:26 EDT 2023
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#10 by Jon E. Schoenfield at Sun Jun 04 21:58:23 EDT 2023
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#9 by Jon E. Schoenfield at Sun Jun 04 21:58:12 EDT 2023
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| FORMULA
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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.
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| EXAMPLE
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(column 1 of A035513) = (1,4,6,9,12,14,17,19,...), so that (column 1 of B(2,1)) = (5,15,26,36,...);
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so that (column 2 of B(2,1)) = (9,25,43,59,...).
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approved
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#8 by N. J. A. Sloane at Fri Apr 07 17:37:34 EDT 2023
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#7 by Jon E. Schoenfield at Thu Apr 06 21:35:38 EDT 2023
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#6 by Jon E. Schoenfield at Thu Apr 06 21:35:28 EDT 2023
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| COMMENTS
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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.
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| FORMULA
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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.
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proposed
editing
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#5 by Jon E. Schoenfield at Thu Apr 06 13:45:49 EDT 2023
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#4 by Jon E. Schoenfield at Thu Apr 06 13:45:31 EDT 2023
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| COMMENTS
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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.
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| EXAMPLE
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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 that (column 1 of B(2,1)) = (5,15,26,36,...),...);
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so that (column 2 of B(2,1)) = (9,25,43,59,...),...).
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proposed
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#3 by Clark Kimberling at Wed Apr 05 12:00:49 EDT 2023
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#2 by Clark Kimberling at Tue Apr 04 19:02:51 EDT 2023
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| NAME
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allocated(2,1)-block array, B(2,1), of the Wythoff array (A035513), read forby Clarkdescending Kimberlingantidiagonals.
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| DATA
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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
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| OFFSET
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1,1
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| COMMENTS
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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.
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| FORMULA
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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.
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| EXAMPLE
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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 that (column 1 of B(2,1)) = (5,15,26,36,...)
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so that (column 2 of B(2,1)) = (9,25,43,59,...)
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| MATHEMATICA
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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 *)
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| CROSSREFS
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Cf. A000045, A001622, A035513, A080164, A361975, A361992 (array B(1,2)), A361994 (array B(2,2)).
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| KEYWORD
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allocated
nonn,tabl
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| AUTHOR
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Clark Kimberling, Apr 04 2023
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| STATUS
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approved
editing
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