A376182 Triangle T read by rows: T(n, k) = (2*n^2 + 4*n + 1 - (-1)^n) / 4 - (1 + (-1)^k) * (n - k) - k.

1, 3, 2, 7, 4, 5, 11, 6, 9, 8, 17, 10, 15, 12, 13, 23, 14, 21, 16, 19, 18, 31, 20, 29, 22, 27, 24, 25, 39, 26, 37, 28, 35, 30, 33, 32, 49, 34, 47, 36, 45, 38, 43, 40, 41, 59, 42, 57, 44, 55, 46, 53, 48, 51, 50, 71, 52, 69, 54, 67, 56, 65, 58, 63, 60, 61, 83, 62, 81, 64, 79, 66, 77, 68, 75, 70, 73, 72

Offset

1,2

Comments

Conjecture: This triangle seen as a sequence yields a permutation of the natural numbers.

Links

Table of n, a(n) for n=1..78.

Formula

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

T(n, k) = T(n, k-2) + 2 * (-1)^k for 3 <= k <= n.

Row sums: Sum_{k=1..n} T(n, k) = (n^3 + n) / 2 + (n - 1) * (1 - (-1)^n) / 4.

G.f.: x*y*(1 + x + x^2*(1 - y)^2 - 3*x^5*y^2 + 2*x^6*y^3 + x^4*y*(4 + y) - x^3*(1 + 4*y + y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 16 2024

From Ruud H.G. van Tol, Sep 22 2024: (Start)

T(n, 1) = A047838(n+1).

T(n, 2) = A033638(n) * 2.

T(n, n) = A000982(n) = (T(n, 1) + T(n, 2) - 1) / 2 for n >= 2. (End)

Example

Triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   3   2
   3 :   7   4   5
   4 :  11   6   9   8
   5 :  17  10  15  12  13
   6 :  23  14  21  16  19  18
   7 :  31  20  29  22  27  24  25
   8 :  39  26  37  28  35  30  33  32
   9 :  49  34  47  36  45  38  43  40  41
  10 :  59  42  57  44  55  46  53  48  51  50
  11 :  71  52  69  54  67  56  65  58  63  60  61
  12 :  83  62  81  64  79  66  77  68  75  70  73  72
  etc.

Prog

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

Crossrefs

Main diagonal is A000982.

Column 1 is A047838(n+1).

Column 2 is 2*A033638.

Cf. A246696 (permutation by row), A246697 (row sums), A376583 (parity).

Sequence in context: A153154 A154438 A354367 * A194071 A194104 A277679

Adjacent sequences: A376179 A376180 A376181 * A376183 A376184 A376185

Keyword

nonn,easy,tabl

Author

Werner Schulte, Sep 14 2024

Status

approved