



9, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744, 633318697598976
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OFFSET

0,1


COMMENTS

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
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
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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4).


FORMULA

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n1), n > 0; a(0)=9.
G.f.: 9/(14*x). (End)
a(n) = 9*A000302(n).  Michel Marcus, Apr 23 2016
E.g.f.: 9*exp(4*x).  Ilya Gutkovskiy, Apr 23 2016
a(n) = 2^(2*n+3) + 2^(2*n).  Andres Cicuttin, Apr 26 2016
a(n) = A004171(n+1) + A000302(n).  Zhandos Mambetaliyev, Nov 19 2016


MATHEMATICA

9*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[4#&, 9, 30] (* Harvey P. Dale, Jan 15 2019 *)


PROG

(MAGMA) [9*4^n: n in [0..30]]; // Vincenzo Librandi, May 19 2011
(PARI) a(n)=9<<n \\ Charles R Greathouse IV, Apr 17 2012


CROSSREFS

Essentially the same as A055841. First differences of A002001.
Cf. A000302.
Sequence in context: A285241 A231431 A264515 * A285674 A075674 A245416
Adjacent sequences: A002060 A002061 A002062 * A002064 A002065 A002066


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



