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 A131577 Zero followed by powers of 2 (cf. A000079). 103
 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 October 6 03:55 EDT 2022. Contains 357261 sequences. (Running on oeis4.)