A048645 - OEIS (original) (raw)
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
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](/A147562 "Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton."), A162795, [A169707](/A169707 "Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighb..."). - Omar E. Pol, Feb 21 2015
Numbers with binary weight 1 or 2. - Omar E. Pol, Feb 22 2015
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
EXAMPLE
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
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
(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
(Python)
from math import isqrt, comb
def A048645(n): return (1<<(m:=isqrt(n-1<<3)+1>>1)-1)+(1<<(n-2-comb(m, 2))) if n>1 else 1 # Chai Wah Wu, Oct 30 2024
CROSSREFS
Cf. A018900, A048623, A046097, [A169707](/A169707 "Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighb..."), [A147562](/A147562 "Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton."), A162795, A003056, A002262, A094373, A028310, A179951.