A099627 Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 23, 31, 32, 33, 35, 39, 47, 63, 64, 65, 67, 71, 79, 95, 127, 128, 129, 131, 135, 143, 159, 191, 255, 256, 257, 259, 263, 271, 287, 319, 383, 511, 512, 513, 515, 519, 527, 543, 575, 639, 767, 1023, 1024, 1025, 1027, 1031, 1039

Offset: 0

Author

Henry Bottomley, Oct 25 2004

Keywords

• easy
• nonn
• tabl

Comments

Positive integers m where m-th Catalan number A000108(m) = C(2m,m)/(m+1) is not divisible by 4, i.e. where A048881(m) is 0 or 1.

Numbers in A000225 or A099628.

From Charles L. Hohn, Jul 25 2024: (Start)

Integers >=1 whose binary digit counts (number of 0s and number of 1s) are distinct from those of any smaller number.

Binary analog of A179239 for n>=1.

All integers whose binary expression conforms to regex /^10*1*$/, shown in base 10 in ascending numeric order. (End)

Together with 0 all fixed points of A073137. - Alois P. Heinz, Jan 30 2025

Examples

			Triangle starts:                  In binary:
   k = 0  1  2  3  4  5
n
0      1                               1
1      2  3                           10     11
2      4  5  7                       100    101    111
3      8  9 11 15                   1000   1001   1011   1111
4     16 17 19 23 31               10000  10001  10011  10111  11111
5     32 33 35 39 47 63           100000 100001 100011 100111 101111 111111
E.g. T(5,3) = 2^5 + 2^3-1 = 32 + 7 = 39 (100111 in binary).
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Crossrefs

A053221 (row sums), A000079 (left diagonal), A000225 (right diagonal).

A048645 (see formula).

Partial sums of A232089.

Cf. A073137, A179239.

Programs

Haskell

a099627 n k = a099627_tabl !! n !! k
a099627_row n = a099627_tabl !! n
a099627_tabl = iterate (\xs@(x:_) -> (2 * x) : map ((+ 1) . (* 2)) xs) [1]
-- Reinhard Zumkeller, Dec 19 2012

Mathematica

Table[2^n+2^k -1,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 27 2016 *)

Formula

As sequence, a(n) = A048645(n+2) - 1.

G.f.: (1 - x - x^2*y)/((1 - x)*(1 - 2*x)*(1 - x*y)*(1 - 2*x*y)). - Stefano Spezia, Aug 11 2024