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%I #14 Jan 18 2022 21:45:22
%S 1,3,2,7,6,4,11,10,8,5,19,18,16,13,9,27,26,24,21,17,12,35,34,32,29,25,
%T 20,14,43,42,40,37,33,28,22,15,59,58,56,53,49,44,38,31,23,75,74,72,69,
%U 65,60,54,47,39,30,91,90,88,85,81,76,70,63,55,46,36,107,106,104,101,97,92,86,79,71,62,52,41
%N Triangle read by rows: T(n,k) = n * 2^e(n) - (4^e(n) - 1) / 3 - k * (k - 1) / 2 with e(n) = 1 + floor(log_2(n)) for n >= 1 and 1 <= k <= n.
%C Conjecture: The terms of the triangle yield a permutation of the positive integers (A000027).
%F T(2^n, 1) = A007583(n) for n >= 0.
%F T(n, 1) - T(n, n) = A000217(n-1) for n > 0.
%F T(n, k) = T(n-1, k) + T(n-1, k-1) - T(n-1-2^(e(n-1)-e(n-2)), k-1) with e(n) = 1 + floor(log_2(n)) for n > 3 and 1 < k < n-1 (conjectured).
%e The triangle T(n, k) for 1 <= k <= n begins:
%e n\k: 1 2 3 4 5 6 7 8 9 10 11
%e ================================================
%e 01 : 1
%e 02 : 3 2
%e 03 : 7 6 4
%e 04 : 11 10 8 5
%e 05 : 19 18 16 13 9
%e 06 : 27 26 24 21 17 12
%e 07 : 35 34 32 29 25 20 14
%e 08 : 43 42 40 37 33 28 22 15
%e 09 : 59 58 56 53 49 44 38 31 23
%e 10 : 75 74 72 69 65 60 54 47 39 30
%e 11 : 91 90 88 85 81 76 70 63 55 46 36
%e etc.
%o (PARI) T(n,k) = my(e=1+logint(n,2)); n*2^e - (4^e-1)/3 - k*(k-1)/2;
%o row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Dec 05 2021
%Y Cf. A000027, A000217, A007583.
%K nonn,easy,tabl
%O 1,2
%A _Werner Schulte_, Dec 04 2021
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