A131094 -id:A131094 - cp's OEIS frontend

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

1, 3, 5, 7, 11, 13, 15, 23, 27, 29, 31, 47, 55, 59, 61, 63, 95, 111, 119, 123, 125, 127, 191, 223, 239, 247, 251, 253, 255, 383, 447, 479, 495, 503, 507, 509, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043

Offset: 1

Reinhard Zumkeller, Mar 06 2003

T(n,n) = A036563(n+1) = 2^(n+1) - 3.

Numbers of the form 2^t - 2^k - 1, 1 <= k < t.

			Triangle begins:
.......... 1 ......... ................ 1
........ 3...5 ....... .............. 11 101
...... 7..11..13 ..... .......... 111 1011 1101
... 15..23..27..29 ... ...... 1111 10111 11011 11101
. 31..47..55..59..61 . . 11111 101111 110111 111011 111101.

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
Cf. A181741 (primes), A208083, subsequence of A089633.
Cf. A131094.

a081118 n k = a081118_tabl !! (n-1) !! (k-1)
a081118_row n = a081118_tabl !! (n-1)
a081118_tabl  = iterate
   (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
a081118_list = concat a081118_tabl
-- Reinhard Zumkeller, Feb 23 2012

Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Apr 09 2020 *)

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

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

A131095 Triangle where n-th row contains the smallest n positive integers (listed in order) with exactly n nonleading 0's in their binary representations and where the smallest term in the n-th row is > that the largest term in the (n-1)th row.

Original entry on oeis.org

2, 4, 9, 17, 18, 20, 33, 34, 36, 40, 65, 66, 68, 72, 80, 129, 130, 132, 136, 144, 160, 257, 258, 260, 264, 272, 288, 320, 513, 514, 516, 520, 528, 544, 576, 640, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 2049, 2050, 2052, 2056, 2064, 2080, 2112, 2176

Offset: 1

Leroy Quet, Jun 14 2007

			Binary representations of the terms in the first few rows:
10
100, 1001
10001, 10010, 10100
100001, 100010, 100100, 101000

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A081118, A131094.

import Data.List (sort, nub)
a131095 n k = a131095_tabl !! (n-1) !! (k-1)
a131095_row n = a131095_tabl !! (n-1)
a131095_tabl = [2] : [4, 9] : [17, 18, 20] : f 4 [17, 18, 20] where
   f v ws = ys : f (v + 1) ys where
     ys = take v $ dropWhile (<= last ws) $ nub $ sort $ concatMap h ws
   h z = [2 * z, 4 * z + 1, 4 * z' + b] where (z', b) = divMod z 2
-- Reinhard Zumkeller, Feb 11 2015

A023416 := proc(n) local brep,i ; brep := convert(n,base,2) ; add( 1-op(i,brep),i=1..nops(brep)) ; end: A131095 := proc(rowmax) local a,r,c ; a := 2 ; for r from 1 to rowmax do c := 1 ; while c <= r do if A023416(a) = r then printf("%d, ",a) ; c := c+1 ; fi ; a := a+1 ; od ; od ; end: A131095(10) ; # R. J. Mathar, Jun 15 2007

More terms from R. J. Mathar, Jun 15 2007

cp's OEIS Frontend

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

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