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

Author

Paul Curtz, Aug 29 2007, Dec 06 2007

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

C

int is (unsigned long n) { return !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

Haskell

a131577 = (`div` 2) . a000079
a131577_list = 0 : a000079_list  -- Reinhard Zumkeller, Dec 09 2012

Magma

[(2^n-0^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011

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 *)

PARI

a(n)=1<Charles R Greathouse IV, Sep 15 2012

Python

def A131577(n): return 1<Chai Wah Wu, Sep 09 2023

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

a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n, 2*k-1). - Taras Goy, Jan 02 2025

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