OFFSET
0,3
COMMENTS
A000079 is the main entry for this sequence.
Binomial transform of A000035.
a(n) = a(n-1)-th even natural numbers (A005846) for n > 1. - Jaroslav Krizek, Apr 25 2009
Where record values greater than 1 occur in A083662: A000045(n)=A083662(a(n)). - Reinhard Zumkeller, Sep 26 2009
Number of compositions of natural number n into parts >0.
The signed sequence 0, 1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, ... is the Lucas U(-2,0) sequence. - R. J. Mathar, Jan 08 2013
In computer programming, these are the only unsigned numbers such that k&(k-1)=0, where & is the bitwise AND operator and numbers are expressed in binary. - Stanislav Sykora, Nov 29 2013
Also the 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 1, 2}. - Robert G. Wilson v, Jul 12 2014
Also the smallest nonnegative superincreasing sequence: each term is larger than the sum of all preceding terms. Indeed, an equivalent definition is a(0)=0, a(n+1)=1+sum_{k=0..n} a(k). - M. F. Hasler, Jan 13 2015
REFERENCES
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2).
FORMULA
a(n) = floor(2^(n-1)). - Robert G. Wilson v, Sep 02 2007
G.f.: x/(1-2*x); a(n) = (2^n-0^n)/2. - Paul Barry, Jan 05 2009
E.g.f.: exp(x)*sinh(x). - Geoffrey Critzer, Oct 28 2012
E.g.f.: x/T(0) where T(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 17 2013
MAPLE
MATHEMATICA
Floor[2^Range[-1, 33]] (* Robert G. Wilson v, Sep 02 2007 *)
Join[{0}, 2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
PROG
(Magma) [(2^n-0^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011
(C) int is (unsigned long n) { return !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */
(PARI) a(n)=1<<n-- \\ Charles R Greathouse IV, Sep 15 2012
(Haskell)
a131577 = (`div` 2) . a000079
a131577_list = 0 : a000079_list -- Reinhard Zumkeller, Dec 09 2012
(Python)
def A131577(n): return 1<<n-1 if n else 0 # Chai Wah Wu, Sep 09 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Aug 29 2007, Dec 06 2007
EXTENSIONS
More terms from Robert G. Wilson v, Sep 02 2007
Edited by N. J. A. Sloane, Sep 13 2007
Edited by M. F. Hasler, Jan 13 2015
STATUS
approved