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A099627
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Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.
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13
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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
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OFFSET
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0,2
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COMMENTS
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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.
Integers >=1 whose binary digit counts (number of 0s and number of 1s) are distinct from those of any smaller number.
All integers whose binary expression conforms to regex /^10*1*$/, shown in base 10 in ascending numeric order. (End)
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LINKS
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FORMULA
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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
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EXAMPLE
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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).
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MATHEMATICA
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Table[2^n+2^k -1, {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Mar 27 2016 *)
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PROG
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(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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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