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a(n) = 3*2^n + 1.
40

%I #54 Sep 19 2024 19:46:25

%S 4,7,13,25,49,97,193,385,769,1537,3073,6145,12289,24577,49153,98305,

%T 196609,393217,786433,1572865,3145729,6291457,12582913,25165825,

%U 50331649,100663297,201326593,402653185,805306369,1610612737,3221225473

%N a(n) = 3*2^n + 1.

%C From _Peter Bala_, Oct 28 2013: (Start)

%C 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.

%C 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) + ....

%C 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)

%C The only squares in this sequence are 4, 25, 49. - _Antti Karttunen_, Sep 24 2023

%H Vincenzo Librandi, <a href="/A181565/b181565.txt">Table of n, a(n) for n = 0..1000</a>

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2005337">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663.

%H S. W. Golomb, <a href="/A004119/a004119.pdf">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2)

%F a(n) = A004119(n+1) = A103204(n+1) for all n >= 0.

%F From _Ilya Gutkovskiy_, Jun 01 2016: (Start)

%F O.g.f.: (4 - 5*x)/((1 - x)*(1 - 2*x)).

%F E.g.f.: (1 + 3*exp(x))*exp(x).

%F a(n) = 3*a(n-1) - 2*a(n-2). (End)

%F a(n) = 2*a(n-1) - 1. - _Miquel Cerda_, Aug 16 2016

%F For n >= 0, A005940(a(n)) = A001248(1+n). - _Antti Karttunen_, Sep 24 2023

%t 3*2^Range[0,50]+1 (* _Vladimir Joseph Stephan Orlovsky_, Mar 24 2011 *)

%t LinearRecurrence[{3,-2},{4,7},40] (* _Harvey P. Dale_, Sep 19 2024 *)

%o (PARI) A181565(n)=3<<n+1

%o (Magma) [3*2^n + 1: n in [0..30]]; // _Vincenzo Librandi_, May 19 2011

%Y Cf. A007283, A020737, A083575, A083683, A083686, A168596, A083705, A195744.

%Y Essentially a duplicate of A004119.

%Y A002253 and A039687 give the primes in this sequence, and A181492 is the subsequence of twin primes.

%K nonn,easy

%O 0,1

%A _M. F. Hasler_, Oct 30 2010