OFFSET
0,3
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.
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
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)
EXAMPLE
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
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*)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Sep 01 2014
EXTENSIONS
Edited by M. F. Hasler, Nov 17 2014
STATUS
approved