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
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
Michael P. Connolly, Probabilistic rounding error analysis for numerical linear algebra, Ph. D. Thesis, Univ. Manchester (UK, 2022). See p. 55.
USA Mathematical Olympiad, Problem 4, 2008.
Eric Weisstein's World of Mathematics, Automatic Set.
Eric Weisstein's World of Mathematics, Binomial Coefficient.
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
As a triangle, 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
Sum_{n>=1} 1/a(n) = 2 + A179951. - Amiram Eldar, Jan 22 2022
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-François 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
(Python)
def ok(n): return 1 <= bin(n)[2:].count('1') <= 2
print([k for k in range(1033) if ok(k)]) # Michael S. Branicky, Jan 22 2022
(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
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 22 2022
CROSSREFS
KEYWORD
AUTHOR
Antti Karttunen, Jul 14 1999
STATUS
approved