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 A004119 a(0)=1; thereafter a(n) = 3*2^(n-1)+1. (Formerly M3308) 18
 1, 4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473, 6442450945, 12884901889 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also Pisot sequence L(4,7) (cf. A008776). Alternatively, define the sequence S(a(1),a(2)) by: a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n > 0. This is S(4,7). a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. E.g., a(2)=7 corresponds to (8,12,16,20,24,28,32). - Augustine O. Munagi, Feb 21 2007 Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3,...]. - Gary W. Adamson, Aug 27 2010 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993 S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy] A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Sequences, 11 (2008), #08.5.4. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (3, -2). FORMULA a(n) = 3a(n-1) - 2a(n-2). For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - Benoit Cloitre, Aug 14 2002 For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - Philippe Deléham, Feb 20 2004 O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)). - R. J. Mathar, Nov 23 2007 For n>0, a(n) = 2*a(n-1)-1. - Vincenzo Librandi, Dec 16 2015 MAPLE A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation MATHEMATICA s=4; lst={1, s}; Do[s=s+(s-1); AppendTo[lst, s], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *) Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *) {1}~Join~LinearRecurrence[{3, -2}, {4, 7}, 33] (* Michael De Vlieger, Dec 16 2015 *) PROG (PARI) a(n)=3<

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)