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 A181565 a(n) = 3*2^n + 1. 37
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS From Peter Bala, Oct 28 2013: (Start) Let x and b be positive real numbers. We define an Engel expansion of x to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the series representation x = b/a(1) + b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) + .... Depending on the values of x and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 we recover the ordinary Engel expansion of x. This sequence gives an Engel expansion of 2/3 to the base 2, with the associated series expansion 2/3 = 2/4 + 2^2/(4*7) + 2^3/(4*7*13) + 2^4/(4*7*13*25) + .... More generally, for n and m positive integers, the sequence [m + 1, n*m + 1, n^2*m + 1,...] gives an Engel expansion of the rational number n/m to the base n. See the cross references for several examples. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 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] Wikipedia, Engel Expansion Index entries for linear recurrences with constant coefficients, signature (3,-2) FORMULA a(n) = A004119(n+1) = A103204(n+1) for all n>=0. From Ilya Gutkovskiy, Jun 01 2016: (Start) O.g.f.: (4 - 5*x)/((1 - x)*(1 - 2*x)). E.g.f.: (1 + 3*exp(x))*exp(x). a(n) = 3*a(n-1) - 2*a(n-2). (End) a(n) = 2*a(n-1) - 1. - Miquel Cerda, Aug 16 2016 MATHEMATICA 3*2^Range[0, 50]+1 (*  Vladimir Joseph Stephan Orlovsky, Mar 24 2011 *) PROG (PARI) A181565(n)=3<

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)