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 A002063 a(n) = 9*4^n. 22
 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 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(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<

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Last modified April 14 07:59 EDT 2021. Contains 342946 sequences. (Running on oeis4.)