%I #93 Sep 08 2022 08:44:29
%S 9,36,144,576,2304,9216,36864,147456,589824,2359296,9437184,37748736,
%T 150994944,603979776,2415919104,9663676416,38654705664,154618822656,
%U 618475290624,2473901162496,9895604649984,39582418599936,158329674399744,633318697598976
%N a(n) = 9*4^n.
%C a(n) is twice the area of the trapezoid created by the four points (2^n,2^(n+1)), (2^(n+1), 2^n), (2^(n+1), 2^(n+2)), (2^(n+2), 2^(n+1)). - _J. M. Bergot_, May 23 2014
%C These are squares that can be expressed as sum of exactly two distinct powers of two. For instance, a(4) = 9*4^4 = 2304 = 2^11 + 2^8 . It is conjectured that these are the only squares with this characteristic (tested on squares up to (10^7)^2). - _Andres Cicuttin_, Apr 23 2016
%C Conjecture is true. It is equivalent to prove that the Diophantine equation m^2 = 2^k*(1+2^h), where h>0, has solutions only when h=3. Dividing by 2^k we must obtain an odd square on the left, since 1+2^h is odd, so we can write (2*r+1)^2 = 1+2^h. Expanding, we have 4*r*(r+1) = 2^h, from which it follows that r must be equal to 1 and thus h=3, since r and r+1 must be powers of 2. - _Giovanni Resta_, Jul 27 2017
%H Vincenzo Librandi, <a href="/A002063/b002063.txt">Table of n, a(n) for n = 0..500</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F From _Philippe Deléham_, Nov 23 2008: (Start)
%F a(n) = 4*a(n-1), n > 0; a(0)=9.
%F G.f.: 9/(1-4*x). (End)
%F a(n) = 9*A000302(n). - _Michel Marcus_, Apr 23 2016
%F E.g.f.: 9*exp(4*x). - _Ilya Gutkovskiy_, Apr 23 2016
%F a(n) = 2^(2*n+3) + 2^(2*n). - _Andres Cicuttin_, Apr 26 2016
%F a(n) = A004171(n+1) + A000302(n). - _Zhandos Mambetaliyev_, Nov 19 2016
%t 9*4^Range[0, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jun 09 2011 *)
%t NestList[4#&,9,30] (* _Harvey P. Dale_, Jan 15 2019 *)
%o (Magma) [9*4^n: n in [0..30]]; // _Vincenzo Librandi_, May 19 2011
%o (PARI) a(n)=9<<n \\ _Charles R Greathouse IV_, Apr 17 2012
%Y Essentially the same as A055841. First differences of A002001.
%Y Cf. A000302.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_