A240857 Triangle read by rows: T(0,0) = 0; T(n+1,k) = T(n,k+1), 0 <= k < n; T(n+1,n) = T(n,0); T(n+1,n+1) = T(n,0)+1.

0, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 2, 3, 0, 1, 1, 2, 2, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 3, 0, 1, 1, 2, 1

Offset: 0

Author

Reinhard Zumkeller, Apr 14 2014

Keywords

• nonn
• tabl

Comments

Let h be the initial term of row n, to get row n+1, remove h and then append h and h+1.

Examples

			.   0:                                 0
.   1:                               0   1
.   2:                             1   0   1
.   3:                           0   1   1   2
.   4:                         1   1   2   0   1
.   5:                       1   2   0   1   1   2
.   6:                     2   0   1   1   2   1   2
.   7:                   0   1   1   2   1   2   2   3
.   8:                 1   1   2   1   2   2   3   0   1
.   9:               1   2   1   2   2   3   0   1   1   2
.  10:             2   1   2   2   3   0   1   1   2   1   2
.  11:           1   2   2   3   0   1   1   2   1   2   2   3
.  12:         2   2   3   0   1   1   2   1   2   2   3   1   2
.  13:       2   3   0   1   1   2   1   2   2   3   1   2   2   3
.  14:     3   0   1   1   2   1   2   2   3   1   2   2   3   2   3
.  15:   0   1   1   2   1   2   2   3   1   2   2   3   2   3   3   4 .
		

Links

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

Crossrefs

Cf. A048881 (left edge), A000120 (right edge), A000788 (row sums), A000523 (row maxima), A240883 (central terms).

Cf. A035327.

Programs

Haskell

a240857 n k = a240857_tabl !! n !! k
a240857_row n = a240857_tabl !! n
a240857_tabl = iterate (\(x:xs) -> xs ++ [x, x + 1]) [0]

Mathematica

T[n_, k_] := DigitCount[n + k + 1, 2, 1] - 1; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2023 *)

Python

from math import isqrt
def A240857(n): return (n-((r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1)))*(r-3)>>1)).bit_count()-1 # Chai Wah Wu, Nov 11 2024

Formula

T(n,k) = A048881(n+k), 0 <= k <= n.

For n > 0: T(n,A035327(n)) = 0.