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A257852
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Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.
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1
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3, 1, 7, 13, 9, 11, 5, 29, 17, 15, 53, 37, 45, 25, 19, 21, 117, 69, 61, 33, 23, 213, 149, 181, 101, 77, 41, 27, 85, 469, 277, 245, 133, 93, 49, 31, 853, 597, 725, 405, 309, 165, 109, 57, 35, 341, 1877, 1109, 981, 533, 373, 197, 125, 65, 39
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OFFSET
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1,1
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COMMENTS
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Sequence is a permutation of the odd natural numbers.
Let N_1 denote the set of odd natural numbers, and let |y|_2 denote 2-adic valuation of y. Define the map F : N_1 -> N_1 by F(x) = (3*x + 1)/2^|3*x+1|_2 (cf. A075677). Then row n of A is the set of all x in N_1 for which |3*x + 1|_2 = n. Hence F(A(n,k)) = 6*k - 3 - 2*(-1)^n.
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LINKS
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FORMULA
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A(n,k+1) = A(n,k) + 2^(n+1).
A(n+2,k) = A(n,k)*4 + 1.
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EXAMPLE
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Array A begins:
n\k| 1| 2| 3| 4| 5| 6| 7| 8| ...
---+---------------------------------------------
1 | 3, 7, 11, 15, 19, 23, 27, 31, ...
2 | 1, 9, 17, 25, 33, 41, 49, 57, ...
3 | 13, 29, 45, 61, 77, 93, 109, 125, ...
4 | 5, 37, 69, 101, 133, 165, 197, 229, ...
5 | 53, 117, 181, 245, 309, 373, 437, 501, ...
6 | 21, 149, 277, 405, 533, 661, 789, 917, ...
... (End)
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MATHEMATICA
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(* Array: *)
Grid[Table[(2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(2^(n - k + 1)*(6*k - 3 - 2*(-1)^(n - k + 1)) - 1)/ 3, {n, 10}, {k, n}]]
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PROG
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(PARI)
up_to = 105;
A257852sq(n, k) = ((2^n * (6*k - 3 - 2*(-1)^n) - 1)/3);
A257852list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A257852sq((a-(col-1)), col))); (v); };
v257852 = A257852list(up_to);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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