

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
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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), 657663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657663. [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(n1)  2*a(n2). (End)
a(n) = 2*a(n1)  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<<n+1
(MAGMA) [3*2^n + 1: n in [0..30]]; // Vincenzo Librandi, May 19 2011


CROSSREFS

Cf. A007283, A020737, A083575, A083683, A083686, A168596, A083705, A195744.
Essentially a duplicate of A004119.
A002253 and A039687 give the primes in this sequence, and A181492 is the subsequence of twin primes.
Sequence in context: A074863 A118334 A205538 * A004119 A074864 A074865
Adjacent sequences: A181562 A181563 A181564 * A181566 A181567 A181568


KEYWORD

nonn,easy


AUTHOR

M. F. Hasler, Oct 30 2010


STATUS

approved



