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Triangle T read by rows: T(n, k) = (n^2 - 4*n - (-1)^n * (n - 4)) / 2 + 4*k - (-1)^n * (1 + (-1)^k).
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%I #6 Oct 16 2024 21:45:12

%S 1,3,5,2,8,10,4,6,12,14,7,13,15,21,23,9,11,17,19,25,27,16,22,24,30,32,

%T 38,40,18,20,26,28,34,36,42,44,29,35,37,43,45,51,53,59,61,31,33,39,41,

%U 47,49,55,57,63,65,46,52,54,60,62,68,70,76,78,84,86,48,50,56,58,64,66,72,74,80,82,88,90

%N Triangle T read by rows: T(n, k) = (n^2 - 4*n - (-1)^n * (n - 4)) / 2 + 4*k - (-1)^n * (1 + (-1)^k).

%C This triangle seen as a sequence yields a permutation of the natural numbers. Next 8*i-5 odd numbers followed by next 8*i-1 even numbers fill next 2 + 2 rows (x, down, x+2, right, x+4, up, x+6, right, x+8, down, x+10, right, x+12, ...).

%F T(n, k) = T(n, k-1) + 4 - (-1)^n * (1 + (-1)^k) for 2 <= k <= n.

%F T(n, k) = T(n, k-2) + 8 for 3 <= k <= n.

%F T(n, n) = (n^2 + 4*n - (-1)^n * (n - 4)) / 2 - (1 + (-1)^n).

%F T(2*n-1, n) = 2 * n^2 - n + 1 + (-1)^n.

%F G.f.: x*y*(1+ 2*x^8*y^4 + x*(2 + 4*y) + x^7*y^3*(3 + 5*y) - x^6*y^2*(8 + 7*y + 2*y^2) - x^2*(3 - y - 3*y^2) + x^4*(6 + 3*y + y^2 + 3*y^3) - x^3*(2 + 9*y + 5*y^2 + 4*y^3) + x^5*y*(1 + y + 5*y^2 - y^3))/((1 - x)^3*(1 + x)^2*(1 - x*y)^3*(1 + x*y)^2). - _Stefano Spezia_, Sep 18 2024

%e Triangle T(n, k) for 1 <= k <= n starts:

%e n \k : 1 2 3 4 5 6 7 8 9 10 11 12

%e ======================================================

%e 1 : 1

%e 2 : 3 5

%e 3 : 2 8 10

%e 4 : 4 6 12 14

%e 5 : 7 13 15 21 23

%e 6 : 9 11 17 19 25 27

%e 7 : 16 22 24 30 32 38 40

%e 8 : 18 20 26 28 34 36 42 44

%e 9 : 29 35 37 43 45 51 53 59 61

%e 10 : 31 33 39 41 47 49 55 57 63 65

%e 11 : 46 52 54 60 62 68 70 76 78 84 86

%e 12 : 48 50 56 58 64 66 72 74 80 82 88 90

%e etc.

%o (PARI) T(n,k)=(n^2-4*n-(-1)^n*(n-4))/2+4*k-(-1)^n*(1+(-1)^k)

%K nonn,easy,tabl

%O 1,2

%A _Werner Schulte_, Sep 17 2024