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%I #52 Sep 24 2023 14:02:48
%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 *)
%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
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