A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046

Offset: 1

Author

Reinhard Zumkeller, Aug 29 2009

Keywords

• nonn
• tabl

Comments

All terms contain exactly 1 zero in binary representation.

Examples

			Initial rows:
   1:                             2
   2:                        5        6
   3:                  11        13        14
   4:             23        27       29        30
   5:        47        55        59        61        62
   6:    95       111       119      123       125       126
also in binary representation:
                                 10
                            101       110
                      1011      1101      1110
                 10111     11011     11101     11110
           101111    110111    111011    111101    111110
      1011111   1101111   1110111   1111011   1111101   1111110 .
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A030130, A023416, A059673, A055010, A086224, A036563, A000918.

Programs

Haskell

a164874 n k = a164874_tabl !! (n-1) !! (k-1)
a164874_row n = a164874_tabl !! (n-1)
a164874_tabl = map reverse $ iterate f [2] where
   f xs@(x:_) = (2 * x + 2) : map ((+ 1) . (* 2)) xs
-- Reinhard Zumkeller, Mar 31 2015

Mathematica

A164874row[n_] := 2^(n + 1) - 1 - BitShiftRight[2^n, Range[n]];
Array[A164874row, 10] (* Paolo Xausa, Jun 13 2025 *)

Python

from math import isqrt
def A164874(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1))-(1<<(a*(a-1)>>1)-n)-1 # Chai Wah Wu, May 21 2025

Formula

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

T(n,k) = A030130(n*(n-1)/2 + k + 1);

A023416(T(n,k)) = 1, 1<=k<=n;

A059673(n) = sum of n-th row;

T(n,1) = A055010(n);

T(n,2) = A086224(n-2) for n > 1;

T(n,n-1) = A036563(n+1) for n > 1;

T(n,n) = A000918(n+1).