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A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.

%I A376133 #11 Sep 13 2024 13:43:13
%S A376133 1,2,4,3,7,5,6,12,8,10,9,17,11,15,13,14,24,16,22,18,20,19,31,21,29,23,
%T A376133 27,25,26,40,28,38,30,36,32,34,33,49,35,47,37,45,39,43,41,42,60,44,58,
%U A376133 46,56,48,54,50,52,51,71,53,69,55,67,57,65,59,63,61,62,84,64,82,66,80,68,78,70,76,72,74
%N A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.
%C A376133 Row n consists of the next n odd/even natural numbers if n is odd/even. So the sequence yields a permutation of the natural numbers.
%F A376133 T(n, k) = (2*n*n + (-1)^k * 4 * (n - k) + 5 + 2 * (-1)^k + (-1)^n) / 4.
%F A376133 T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 = A061925(n-1).
%F A376133 T(n, 2) = (2*n*n + 4*n - 1 + (-1)^n) / 4 = A074148(n) for n > 1.
%F A376133 T(n, k) = T(n, k-2) - (-1)^k * 2 for 3 <= k <= n.
%F A376133 G.f.: x*y*(1 + 2*x*y + 2*x^5*y^2 + x^6*y^3 - x^4*y*(3 + y + y^2) - x^2*(1 + y + 3*y^2) + 2*x^3*(1 + y^3))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - _Stefano Spezia_, Sep 12 2024
%e A376133 Row n=5: Next (1,3,5,7 see rows 1 and 3) five odd numbers are 9,11,13,15 and 17; with "9+8-6+4-2" we get 9,17,11,15,13 for row 5.
%e A376133 Row n=8: Next (2,4,..,24 see rows 2, 4 and 6) eight even numbers are 26,28,..,40; with "26+14-12+10-8+6-4+2" we get 26,40,28,38,30,36,32,34 for row 8.
%e A376133 Triangle T(n, k) for 1 <= k <= n starts:
%e A376133 n\ k :   1   2   3   4   5   6   7   8   9  10  11  12
%e A376133 ======================================================
%e A376133    1 :   1
%e A376133    2 :   2   4
%e A376133    3 :   3   7   5
%e A376133    4 :   6  12   8  10
%e A376133    5 :   9  17  11  15  13
%e A376133    6 :  14  24  16  22  18  20
%e A376133    7 :  19  31  21  29  23  27  25
%e A376133    8 :  26  40  28  38  30  36  32  34
%e A376133    9 :  33  49  35  47  37  45  39  43  41
%e A376133   10 :  42  60  44  58  46  56  48  54  50  52
%e A376133   11 :  51  71  53  69  55  67  57  65  59  63  61
%e A376133   12 :  62  84  64  82  66  80  68  78  70  76  72  74
%e A376133   etc.
%p A376133 T := (n, k) -> ((-1)^k*(2 + 4*(n - k)) + 2*n^2 + (-1)^n + 5)/4:
%p A376133 seq(seq(T(n, k), k = 1..n), n = 1..12);  # _Peter Luschny_, Sep 13 2024
%o A376133 (PARI) T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4
%Y A376133 Cf. A061925 (column 1), A074148 (column 2), A074149 (row sums), A236283 (main diagonal).
%K A376133 nonn,easy,tabl
%O A376133 1,2
%A A376133 _Werner Schulte_, Sep 11 2024