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

0,2

Comments

Compare this triangle to A364390.

Links

Table of n, a(n) for n=0..77.

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

Cf. A006003, A042948, A364390.

Sequence in context: A328258 A190415 A024602 * A131603 A242554 A219176

Adjacent sequences: A367266 A367267 A367268 * A367270 A367271 A367272

Keyword

nonn,easy,tabl

Author

Werner Schulte, Dec 06 2023

Status

approved