A367269 Triangle T(n, k) read by rows and based on A042948 yields a permutation of the natural numbers.
1, 4, 3, 6, 5, 2, 13, 12, 9, 8, 15, 14, 11, 10, 7, 26, 25, 22, 21, 18, 17, 28, 27, 24, 23, 20, 19, 16, 43, 42, 39, 38, 35, 34, 31, 30, 45, 44, 41, 40, 37, 36, 33, 32, 29, 64, 63, 60, 59, 56, 55, 52, 51, 48, 47, 66, 65, 62, 61, 58, 57, 54, 53, 50, 49, 46, 89, 88, 85, 84, 81, 80, 77, 76, 73, 72, 69, 68
Offset: 0
Examples
Triangle T(n, k) for 0 <= k <= n starts: n\k : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ================================================================= 0 : 1 1 : 4 3 2 : 6 5 2 3 : 13 12 9 8 4 : 15 14 11 10 7 5 : 26 25 22 21 18 17 6 : 28 27 24 23 20 19 16 7 : 43 42 39 38 35 34 31 30 8 : 45 44 41 40 37 36 33 32 29 9 : 64 63 60 59 56 55 52 51 48 47 10 : 66 65 62 61 58 57 54 53 50 49 46 11 : 89 88 85 84 81 80 77 76 73 72 69 68 12 : 91 90 87 86 83 82 79 78 75 74 71 70 67 13 : 118 117 114 113 110 109 106 105 102 101 98 97 94 93 14 : 120 119 116 115 112 111 108 107 104 103 100 99 96 95 92 etc.
Programs
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Mathematica
T[n_, k_]:= (n+1) * (n+2) / 2 + n * Mod[n,2] - 2 * k + Mod[k,2]; Table[T[n,k],{n,0,11},{k,0,n}]//Flatten (* Stefano Spezia, Dec 06 2023 *) -
PARI
T(n, k) = (n+1)*(n+2)/2+n*(n%2)-2*k+(k%2)
Formula
T(n, k) = (n+1) * (n+2) / 2 + n * (n mod 2) - 2 * k + (k mod 2) for 0 <= k <= n.
T(n, k) = T(n, 0) + A042948(k) for 0 <= k <= n.
T(n, 0) = (n+1) * (n+2) / 2 + n * (n mod 2) for n >= 0.
T(n, n) = (n^2 - n + 2) / 2 + (n+1) * (n mod 2) for n >= 0.
T(2*n, n) = 2 * n^2 + n + 1 + (n mod 2) for n >= 0.
T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-1) for 0 < k < n.
Row sums: A006003(n+1) - 2 * (-1)^n * (floor((n+1)/2))^2 for n >= 0.
G.f. of column k = 0: F(t, 0) = Sum_{n>=0} T(n, 0) * t^n = (1 + 3*t + t^3 - t^4) / ((1-t)^3 * (1+t)^2).
G.f.: F(t, x) = Sum_{n>=0, k=0..n} T(n, k) * x^k * t^n = (F(t, 0) - x * F(x*t, 0)) / (1-x) - 2*x*t / ((1-t) * (1-x*t)^2) + x*t / ((1-t) * (1-x^2*t^2)).
Alt. row sums: (n^(2 - n mod 2) + 2 - n mod 2) / 2 for n >= 0.
Comments