VOOZH about

URL: https://oeis.org/A376182

⇱ A376182 - OEIS

login
A376182
Triangle T read by rows: T(n, k) = (2*n^2 + 4*n + 1 - (-1)^n) / 4 - (1 + (-1)^k) * (n - k) - k.
0
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.
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
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Sep 14 2024
STATUS
approved