cp's OEIS Frontend

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

%I A131577 #100 Jan 03 2025 09:34:30
%S A131577 0,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A131577 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,
%U A131577 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592
%N A131577 Zero followed by powers of 2 (cf. A000079).
%C A131577 A000079 is the main entry for this sequence.
%C A131577 Binomial transform of A000035.
%C A131577 Essentially the same as A034008 and A000079.
%C A131577 a(n) = a(n-1)-th even natural numbers (A005846) for n > 1. - _Jaroslav Krizek_, Apr 25 2009
%C A131577 Where record values greater than 1 occur in A083662: A000045(n)=A083662(a(n)). - _Reinhard Zumkeller_, Sep 26 2009
%C A131577 Number of compositions of natural number n into parts >0.
%C A131577 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
%C A131577 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
%C A131577 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
%C A131577 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
%D A131577 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
%H A131577 Vincenzo Librandi, <a href="/A131577/b131577.txt">Table of n, a(n) for n = 0..1000</a>
%H A131577 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani/Compositions_of_natural_numbers_over_arithmetic_progressions"> Compositions of natural numbers over arithmetic progressions </a>
%H A131577 Jimmy Devillet, <a href="https://arxiv.org/abs/1712.07856">Bisymmetric and quasitrivial operations: characterizations and enumerations</a>, arXiv:1712.07856 [math.RA], 2017.
%H A131577 J. T. Rowell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Rowell/rowell3.html">Solution Sequences for the Keyboard Problem and its Generalizations</a>, Journal of Integer Sequences, 18 (2015), #15.10.7.
%H A131577 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H A131577 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%H A131577 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F A131577 a(n) = floor(2^(n-1)). - _Robert G. Wilson v_, Sep 02 2007
%F A131577 G.f.: x/(1-2*x); a(n) = (2^n-0^n)/2. - _Paul Barry_, Jan 05 2009
%F A131577 E.g.f.: exp(x)*sinh(x). - _Geoffrey Critzer_, Oct 28 2012
%F A131577 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
%F A131577 a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n, 2*k-1). - _Taras Goy_, Jan 02 2025
%p A131577 A131577 := proc(n)
%p A131577     if n =0 then
%p A131577         0;
%p A131577     else
%p A131577         2^(n-1) ;
%p A131577     end if;
%p A131577 end proc: # _R. J. Mathar_, Jul 22 2012
%t A131577 Floor[2^Range[-1, 33]] (* _Robert G. Wilson v_, Sep 02 2007 *)
%t A131577 Join[{0}, 2^Range[0, 60]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 09 2011 *)
%o A131577 (Magma) [(2^n-0^n)/2: n in [0..50]]; // _Vincenzo Librandi_, Aug 10 2011
%o A131577 (C) int is (unsigned long n) { return !(n & (n-1)); } /* _Charles R Greathouse IV_, Sep 15 2012 */
%o A131577 (PARI) a(n)=1<<n-- \\ _Charles R Greathouse IV_, Sep 15 2012
%o A131577 (Haskell)
%o A131577 a131577 = (`div` 2) . a000079
%o A131577 a131577_list = 0 : a000079_list  -- _Reinhard Zumkeller_, Dec 09 2012
%o A131577 (Python)
%o A131577 def A131577(n): return 1<<n-1 if n else 0 # _Chai Wah Wu_, Sep 09 2023
%Y A131577 Cf. A000079, A003945, A042950, A020406, A046045, A011782.
%K A131577 nonn,easy
%O A131577 0,3
%A A131577 _Paul Curtz_, Aug 29 2007, Dec 06 2007
%E A131577 More terms from _Robert G. Wilson v_, Sep 02 2007
%E A131577 Edited by _N. J. A. Sloane_, Sep 13 2007
%E A131577 Edited by _M. F. Hasler_, Jan 13 2015