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A048645
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Integers with one or two 1-bits in their binary expansion.
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32
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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
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table;
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refs;
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history;
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OFFSET
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1,2
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COMMENTS
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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
Also numbers n such that n!/2^(n-2) is an integer. - Michel Lagneau, Mar 28 2011
Numbers with binary weight 1 or 2. - Omar E. Pol, Feb 22 2015
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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PROG
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(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
(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
(Python)
def ok(n): return 1 <= bin(n)[2:].count('1') <= 2
(Python)
from itertools import count, islice
def agen(): # generator of terms
for d in count(0):
msb = 2**d
yield msb
for lsb in range(d):
yield msb + 2**lsb
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CROSSREFS
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Cf. A018900, A048623, A046097, A169707, A147562, A162795, A003056, A002262, A094373, A028310, A179951.
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KEYWORD
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AUTHOR
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STATUS
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approved
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