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A048645 Integers with one or two 1-bits in their binary expansion. 30
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Apart from initial 1, sums of two not necessarily distinct powers of 2.

4 does not divide C(2s-1,s) (= A001700[ s ]) if and only if s=a(n).

Possible number of sides of a regular polygon such that there exists a triangulation where each triangle is isosceles. - Sen-peng Eu, May 07 2008

For n > 1, 1 < k <= n: T(n,1) = A173786(n-2,n-2) and T(n,k) = A173786(n-1,k-2). - Reinhard Zumkeller, Feb 28 2010

Also numbers n such that n!/2^(n-2) is an integer. - Michel Lagneau, Mar 28 2011

It appears these are also the indices of the terms that are shared by the cellular automata of A147562, A162795, A169707. - Omar E. Pol, Feb 21 2015

Numbers with binary weight 1 or 2. - Omar E. Pol, Feb 22 2015

LINKS

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

USA Mathematical Olympiad, Problem 4, 2008.

Eric Weisstein's World of Mathematics, Automatic Set

Eric Weisstein's World of Mathematics, Binomial Coefficient

Index entries for sequences related to cellular automata

FORMULA

a(0) = 1, a(n) = (2^(trinv(n-1)-1) + 2^((n-1)-((trinv(n-1)*(trinv(n-1)-1))/2))), i.e., 2^A003056(n) + 2^A002262(n-1) (the latter sequence contains the definition of trinv).

Let Theta = Sum_{k >= 0} x^(2^k). Then Sum_{n>=1} x^a(n) = (Theta^2 + Theta + x)/2. - N. J. A. Sloane, Jun 23 2009

It appears that A147562(a(n)) = A162795(a(n)) = A169707(a(n)). - Omar E. Pol, Feb 19 2015

EXAMPLE

From Omar E. Pol, Feb 18 2015: (Start)

Also, written as a triangle T(j,k), k >= 1, in which row lengths are the terms of A028310:

   1;

   2;

   3,  4;

   5,  6,  8;

   9, 10, 12, 16;

  17, 18, 20, 24, 32;

  33, 34, 36, 40, 48, 64;

  65, 66, 68, 72, 80, 96, 128;

  ...

It appears that column 1 is A094373.

It appears that the right border gives A000079.

It appears that the first differences in every row that contains at least two terms give the first h-1 powers of 2, where h is the length of the row.

(End)

MAPLE

lincom:=proc(a, b, n) local i, j, s, m; s:={}; for i from 0 to n do for j from 0 to n do m:=a^i+b^j; if m<=n then s:={op(s), m} fi od; od; lprint(sort([op(s)])); end: lincom(2, 2, 1000); # Zerinvary Lajos, Feb 24 2007

MATHEMATICA

Select[Range[2000], 1 <= DigitCount[#, 2, 1] <= 2&] (* Jean-Fran├žois Alcover, Mar 06 2016 *)

PROG

(Haskell)

import Data.List (insert)

a048645 n k = a048645_tabl !! (n-1) !! (k-1)

a048645_row n = a048645_tabl !! (n-1)

a048645_tabl = iterate (\xs -> insert (2 * head xs + 1) $ map ((* 2)) xs) [1]

a048645_list = concat a048645_tabl

-- Reinhard Zumkeller, Dec 19 2012

(PARI) isok(n) = my(hw = hammingweight(n)); (hw == 1) || (hw == 2); \\ Michel Marcus, Mar 06 2016

(PARI) a(n) = if(n <= 2, return(n), n-=2); my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)) \\ David A. Corneth, Jan 02 2019

(PARI) nxt(n) = msb = 1 << logint(n, 2); if(n == msb, n + 1, t = n - msb; n + t) \\ David A. Corneth, Jan 02 2019

CROSSREFS

Cf. A018900, A048623, A046097, A169707, A147562, A162795, A003056, A002262, A094373, A028310.

Sequence in context: A018412 A061945 A029509 * A173786 A093863 A091902

Adjacent sequences:  A048642 A048643 A048644 * A048646 A048647 A048648

KEYWORD

easy,nonn,base,tabl,changed

AUTHOR

Antti Karttunen, Jul 14 1999

STATUS

approved

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Last modified January 18 00:34 EST 2019. Contains 319255 sequences. (Running on oeis4.)