A246694 Triangle read by rows: T(n,k) = T(n,k-2) + 1 if n > 1 and 2 <= k <= n; T(0,0) = 1, T(1,0) = 1, T(1,1) = 2; if n > 1 is odd, then T(n,0) = T(n-1,n-2) + 1 and T(n,1) = T(n-1,n-1) + 1; if n > 1 is even, then T(n,0) = T(n-1,n-1) + 1 and T(n,1) = T(n-1,n-2) + 1.

1, 1, 2, 3, 2, 4, 3, 5, 4, 6, 7, 5, 8, 6, 9, 7, 10, 8, 11, 9, 12, 13, 10, 14, 11, 15, 12, 16, 13, 17, 14, 18, 15, 19, 16, 20, 21, 17, 22, 18, 23, 19, 24, 20, 25, 21, 26, 22, 27, 23, 28, 24, 29, 25, 30, 31, 26, 32, 27, 33, 28, 34, 29, 35, 30, 36, 31, 37, 32

Offset: 0

Author

Clark Kimberling, Sep 01 2014

Keywords

• nonn
• easy
• tabl

Comments

As an array, for each m, row 2*m has m odd numbers and m+1 even numbers; row 2*m-1 has m odds and m evens. As a sequence, every positive integer n occurs exactly twice, separated by floor((n+1)/2) other numbers.

Examples

			First 8 rows:
  1
  1 ... 2
  3 ... 2 ... 4
  3 ... 5 ... 4 ... 6
  7 ... 5 ... 8 ... 6 ... 9
  7 .. 10 ... 8 .. 11 ... 9 .. 12
 13 .. 10 .. 14 .. 11 .. 15 .. 12 .. 16
 13 .. 17 .. 14 .. 18 .. 15 .. 19 .. 16 .. 20
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Crossrefs

Cf. A246705, A246706, A246696.

Cf. A246695 (row sums), A174114 (central terms).

Cf. A002620 (main diagonal and first subdiagonal), A377802.

Programs

Haskell

a246694 n k = a246694_tabl !! n !! k
a246694_row n = a246694_tabl !! n
a246694_tabl = [1] : [1,2] : f 1 2 [1,2] where
   f i z xs = ys : f j (z + 1) ys where
     ys = take (z + 1) $ map (+ 1) (xs !! (z - i) : xs !! (z - j) : ys)
     j = 3 - i
-- Reinhard Zumkeller, Sep 03 2014

Mathematica

z = 25; t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 2;
t[n_, 0] := If[OddQ[n], t[n - 1, n - 2] + 1, t[n - 1, n - 1] + 1];
t[n_, 1] := If[OddQ[n], t[n - 1, n - 1] + 1, t[n - 1, n - 2] + 1];
t[n_, k_] := t[n, k - 2] + 1; Flatten[Table[t[n, k], {n, 0, z}, {k, 0, n}]](*A246694*)

Formula

T(n, k) = 1 + floor(n/2) * (1+(-1)^k) / 2 + (floor(n/2))^2 + (2*k - 1 + (-1)^k) / 4 + (1-(-1)^n) * (1-(-1)^k) * n / 4. - Werner Schulte, Nov 16 2024

From Stefano Spezia, Nov 17 2024: (Start)

T(n, k) = (6 + (-1)^k + (-1)^(k+n) + 4*k + 2*n*(1 + (-1)^(k+n) + n))/8.

G.f.: (1 - x^3*y + x^7*y^3 + x^4*(1 - 2*y^2) - x^5*y*(1 - y^2))/((1 - x)^3*(1 + x)^2*(1 - x*y)^3*(1 + x*y)). (End)

Extensions

Edited by M. F. Hasler, Nov 17 2014