A131577 Zero followed by powers of 2 (cf. A000079).

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset

0,3

Comments

A000079 is the main entry for this sequence.

Binomial transform of A000035.

Essentially the same as A034008 and A000079.

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

Adi Dani, Compositions of natural numbers over arithmetic progressions

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).

Index entries for Lucas sequences

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

A131577 := proc(n)
    if n =0 then
        0;
    else
        2^(n-1) ;
    end if;
end proc: # R. J. Mathar, Jul 22 2012

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

Cf. A000079, A003945, A042950, A020406, A046045, A011782.

Sequence in context: A000079 A011782 A120617 * A155559 A166444 A171449

Adjacent sequences: A131574 A131575 A131576 * A131578 A131579 A131580

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