A002063 - OEIS

A002063

a(n) = 9*4^n.

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(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