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 A131577 Zero followed by powers of 2 (cf. A000079). 74
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 Floor(2^(k-1)) with k=-1..n. - Robert G. Wilson v 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 *) 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<

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Last modified August 19 20:20 EDT 2019. Contains 326133 sequences. (Running on oeis4.)