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A023758
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Numbers of the form 2^i - 2^j with i >= j.
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24
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0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 96, 112, 120, 124, 126, 127, 128, 192, 224, 240, 248, 252, 254, 255, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1023
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OFFSET
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1,3
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COMMENTS
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Numbers whose digits in base 2 are in nonincreasing order.
Might be called "nialpdromes".
Subset of A077436. Proof: Since a(n) is of form (2^i-1)2^j, i,j>=0, a(n)^2 = [2^(2i)-2^(i+1)]2^(2j) + 2^(2j) where the first sum term has i-1 one bits and its 2j-th bit is zero, while the second sum term switches the 2j-th bit to one, giving i one bits, as in a(n). - Ralf Stephan, Mar 08 2004
Numbers n such that binary representation contains no "01". - Benoit Cloitre, May 23 2004
Every polynomial with coefficients equal to 1 for the leading terms and 0 after that, evaluated at 2. For instance a(13) = x^4 + x^3 + x^2 at 2, a(14) = x^4 + x^3 + x^2 + x at 2. - Ben Paul Thurston, Jan 11 2008
From Gary W. Adamson, Jul 18 2008: (Start)
As a triangle by rows starting:
1;
2, 3;
4, 6, 7;
8, 12, 14, 15;
16, 24, 28, 30, 31;
...,
equals A000012 * A130123 * A000012, where A130123 = (1, 0,2; 0,0,4; 0,0,0,8;...). Row sums of this triangle = A000337 starting (1, 5, 17, 49, 129,...). (End)
First differences are A057728 = 1; 1; 1; 1; 2,1; 1; 4,2,1; 1; 8,4,2,1; 1;... i.e., decreasing powers of 2, separated by another "1". - M. F. Hasler, May 06 2009
Apart from first term, numbers that are powers of 2 or the sum of some consecutive powers of 2. - Omar E. Pol, Feb 14 2013
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LINKS
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T. D. Noe and R. Zumkeller, Table of n, a(n) for n = 1..10000, First 5051 terms by T. D. Noe
S. M. Shabab Hossain, Md. Mahmudur Rahman and M. Sohel Rahman, Solving a Generalized Version of the Exact Cover Problem with a Light-Based Device, Optical Supercomputing, Lecture Notes in Computer Science, 2011, Volume 6748/2011, 23-31, DOI: 10.1007/978-3-642-22494-2_4.
Eric Weisstein's World of Mathematics, Digit
Index entries for sequences related to binary expansion of n
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FORMULA
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a(n) = 2^s(n) - 2^{[s(n)^2+s(n)-2n]/2} where s(n) = ceiling{[ -1+sqrt(1+8n)]/2}. - Sam Alexander, Jan 08 2005
a(n) = 2^k + a(n-k-1) for 1 < n and k = A003056(n-2). The rows of T(r, c) = 2^r-2^c for 0 <= c < r read from right to left produce this sequence: 1; 2, 3; 4, 6, 7; 8, 12, 14, 15; ... - Frank Ellermann, Dec 06 2001
For n>0, a(n) mod 2 == A010054(n). - Benoit Cloitre, May 23 2004
A140130(a(n)) = 1 and for n>1: A140129(a(n)) = A002262(n-2). - Reinhard Zumkeller, May 14 2008
a(n+1) = (2^(n-r(r-1)/2)-1) 2^(r(r+1)/2-n), where r=round(sqrt(2n)). - M. F. Hasler, May 06 2009
Start with A000225. If n is in sequence, then so is 2n. - Ralf Stephan, Aug 16 2013
G.f.: x^2/((2-x)*(1-x))*(1+sum(k >=0, x^((k^2+k)/2)*(1 + x*(2^k-1)))). The sum is related to Jacobi theta functions. - Robert Israel, Feb 24 2015
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EXAMPLE
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a(22) = 64 = 32 + 32 = 2^5 + a(16) = 2^A003056(20) + a(22-5-1).
a(23) = 96 = 64 + 32 = 2^6 + a(16) = 2^A003056(21) + a(23-6-1).
a(24) = 112 = 64 + 48 = 2^6 + a(17) = 2^A003056(22) + a(24-6-1).
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MAPLE
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a:=proc(n) local n2, d: n2:=convert(n, base, 2): d:={seq(n2[j]-n2[j-1], j=2..nops(n2))}: if n=0 then 0 elif n=1 then 1 elif d={0, 1} or d={0} or d={1} then n else fi end: seq(a(n), n=0..2100); # Emeric Deutsch, Apr 22 2006
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MATHEMATICA
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Union[Flatten[Table[2^i - 2^j, {i, 0, 100}, {j, 0, i}]]] (* T. D. Noe, Mar 15 2011 *)
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PROG
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(PARI) for(n=0, 2500, if(prod(k=1, length(binary(n))-1, component(binary(n), k)+1-component(binary(n), k+1))>0, print1(n, ", ")))
(PARI) A023758(n)={ my(r=round(sqrt(2*n--))); (1<<(n-r*(r-1)/2)-1)<<(r*(r+1)/2-n) }
/* or, to illustrate the "decreasing digit" property and analogy to A064222: */
A023758(n, show=0)={ my(a=0); while(n--, show & print1(a", "); a=vecsort(binary(a+1)); a*=vector(#a, j, 2^(j-1))~); a} \\ M. F. Hasler, May 06 2009
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a023758 n = a023758_list !! (n-1)
a023758_list = 0 : f (singleton 1) where
f s = x : f (if even x then insert z s' else insert z $ insert (z+1) s')
where z = 2*x; (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 24 2014, Dec 19 2012
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CROSSREFS
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A000337(r) = sum of row T(r, c) with 0 <= c < r. See also A003056.
Cf. A130123, A000337.
This is the base-2 version of A064222. First differences are A057728. - M. F. Hasler, May 06 2009
Cf. A175332, A007088.
Sequence in context: A114391 A077436 A082752 * A054784 A018585 A018399
Adjacent sequences: A023755 A023756 A023757 * A023759 A023760 A023761
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Olivier Gérard
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EXTENSIONS
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Definition changed by N. J. A. Sloane, Jan 05 2008
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STATUS
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approved
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