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Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.
13

%I #35 Aug 11 2024 23:38:22

%S 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,

%T 79,95,127,128,129,131,135,143,159,191,255,256,257,259,263,271,287,

%U 319,383,511,512,513,515,519,527,543,575,639,767,1023,1024,1025,1027,1031,1039

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

%C 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.

%C Numbers in A000225 or A099628.

%C From _Charles L. Hohn_, Jul 25 2024: (Start)

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

%C Binary analog of A179239 for n>=1.

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

%H Reinhard Zumkeller, <a href="/A099627/b099627.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F G.f.: (1 - x - x^2*y)/((1 - x)*(1 - 2*x)*(1 - x*y)*(1 - 2*x*y)). - _Stefano Spezia_, Aug 11 2024

%e Triangle starts: In binary:

%e k = 0 1 2 3 4 5

%e n

%e 0 1 1

%e 1 2 3 10 11

%e 2 4 5 7 100 101 111

%e 3 8 9 11 15 1000 1001 1011 1111

%e 4 16 17 19 23 31 10000 10001 10011 10111 11111

%e 5 32 33 35 39 47 63 100000 100001 100011 100111 101111 111111

%e E.g. T(5,3) = 2^5 + 2^3-1 = 32 + 7 = 39 (100111 in binary).

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

%o (Haskell)

%o a099627 n k = a099627_tabl !! n !! k

%o a099627_row n = a099627_tabl !! n

%o a099627_tabl = iterate (\xs@(x:_) -> (2 * x) : map ((+ 1) . (* 2)) xs) [1]

%o -- _Reinhard Zumkeller_, Dec 19 2012

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

%Y A048645 (see formula).

%Y Partial sums of A232089.

%Y Cf. A179239.

%K easy,nonn,tabl

%O 0,2

%A _Henry Bottomley_, Oct 25 2004