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A002063 a(n) = 9*4^n. 25
9, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744, 633318697598976 (list; graph; refs; listen; history; text; internal format)
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
Tanya Khovanova, Recursive Sequences
FORMULA
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1), n > 0; a(0)=9.
G.f.: 9/(1-4*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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)