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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Essentially the same as A055841. First differences of A002001.

Cf. A000302.

Programs

Magma

[9*4^n: n in [0..30]]; // Vincenzo Librandi, May 19 2011

Mathematica

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

PARI

a(n)=9<Charles R Greathouse IV, Apr 17 2012

Python

def A002063(n): return 9<<(n<<1) # Chai Wah Wu, Apr 09 2025

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