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A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

%I A081118 #23 Jul 04 2020 02:21:40
%S A081118 1,3,5,7,11,13,15,23,27,29,31,47,55,59,61,63,95,111,119,123,125,127,
%T A081118 191,223,239,247,251,253,255,383,447,479,495,503,507,509,511,767,895,
%U A081118 959,991,1007,1015,1019,1021,1023,1535,1791,1919,1983,2015,2031,2039,2043
%N A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.
%C A081118 T(n,n) = A036563(n+1) = 2^(n+1) - 3.
%C A081118 Numbers of the form 2^t - 2^k - 1, 1 <= k < t.
%H A081118 Reinhard Zumkeller, <a href="/A081118/b081118.txt">Rows n=1..150 of triangle, flattened</a>
%F A081118 T(n, k) = 2^(n+1) - 2^(n-k+1) - 1, 1<=k<=n.
%F A081118 a(n) = (2^A002260(n)-1)*2^A004736(n)-1; a(n)=(2^i-1)*2^j-1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Apr 04 2013
%e A081118 Triangle begins:
%e A081118 .......... 1 ......... ................ 1
%e A081118 ........ 3...5 ....... .............. 11 101
%e A081118 ...... 7..11..13 ..... .......... 111 1011 1101
%e A081118 ... 15..23..27..29 ... ...... 1111 10111 11011 11101
%e A081118 . 31..47..55..59..61 . . 11111 101111 110111 111011 111101.
%t A081118 Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* _Harvey P. Dale_, Apr 09 2020 *)
%o A081118 (Haskell)
%o A081118 a081118 n k = a081118_tabl !! (n-1) !! (k-1)
%o A081118 a081118_row n = a081118_tabl !! (n-1)
%o A081118 a081118_tabl  = iterate
%o A081118    (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
%o A081118 a081118_list = concat a081118_tabl
%o A081118 -- _Reinhard Zumkeller_, Feb 23 2012
%Y A081118 Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
%Y A081118 Cf. A181741 (primes), A208083, subsequence of A089633.
%Y A081118 Cf. A131094.
%K A081118 nonn,tabl
%O A081118 1,2
%A A081118 _Reinhard Zumkeller_, Mar 06 2003