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%I #13 Nov 06 2015 13:32:13
%S 1,4,3,2,8,5,14,10,12,7,6,30,18,16,9,54,38,46,26,20,11,22,118,70,62,
%T 34,24,13,214,150,182,102,78,42,28,15,86,470,278,246,134,94,50,32,17,
%U 854,598,726,406,310,166,110,58,36,19
%N Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2 + 2^(n-1)*(6*k - 3 + 2*(-1)^n))/3, n,k >= 1.
%C The sequence is a permutation of the natural numbers.
%C Theorem: Let v(y) denote the 2-adic valuation of y. For x an odd natural number, let F(x) = (3*x+1)/2^v(3*x+1) (see A075677). Row n of A is the set of all natural numbers m such that v(1+F(4*(2*m-1)-3)) = n.
%F A(n,k) = (1 + A257499(n,k))/2.
%e Array begins:
%e . 1 3 5 7 9 11 13 15 17 19
%e . 4 8 12 16 20 24 28 32 36 40
%e . 2 10 18 26 34 42 50 58 66 74
%e . 14 30 46 62 78 94 110 126 142 158
%e . 6 38 70 102 134 166 198 230 262 294
%e . 54 118 182 246 310 374 438 502 566 630
%e . 22 150 278 406 534 662 790 918 1046 1174
%e . 214 470 726 982 1238 1494 1750 2006 2262 2518
%e . 86 598 1110 1622 2134 2646 3158 3670 4182 4694
%e . 854 1878 2902 3926 4950 5974 6998 8022 9046 10070
%t (* Array: *)
%t Grid[Table[(2 + 2^(n - 1)*(6*k - 3 + 2*(-1)^n))/3, {n, 10}, {k, 10}]]
%t (* Array antidiagonals flattened: *)
%t Flatten[Table[(2 + 2^(n - k)*(6*k - 3 + 2*(-1)^(n - k + 1)))/3, {n, 10}, {k, n}]]
%Y Cf. A005408, A008586, A017089 (rows 1-3).
%Y Cf. A075677, A257480, A257499.
%K nonn,tabl
%O 1,2
%A _L. Edson Jeffery_, May 29 2015
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