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A367269
Triangle T(n, k) read by rows and based on A042948 yields a permutation of the natural numbers.
0
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
0,2
COMMENTS
Compare this triangle to A364390.
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.
EXAMPLE
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.
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 *)
PROG
(PARI) T(n, k) = (n+1)*(n+2)/2+n*(n%2)-2*k+(k%2)
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Dec 06 2023
STATUS
approved