A016802 a(n) = (4*n)^2.

0, 16, 64, 144, 256, 400, 576, 784, 1024, 1296, 1600, 1936, 2304, 2704, 3136, 3600, 4096, 4624, 5184, 5776, 6400, 7056, 7744, 8464, 9216, 10000, 10816, 11664, 12544, 13456, 14400, 15376, 16384, 17424, 18496, 19600, 20736, 21904, 23104, 24336, 25600, 26896, 28224

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A bisection of A016742. Sequence arises from reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Also, sequence found by reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000 (first 200 terms from Ivan Panchenko).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A001105, A001539, A016742, A016754, A016814, A016826, A016838, A112628.

Cf. A008598, A074377, A139098, A195146.

Programs

PARI

a(n) = (4*n)^2; \\ Michel Marcus, Mar 04 2014

Python

def A016802(n): return (4*n)**2 # Karl-Heinz Hofmann, Sep 11 2024

Formula

a(n) = 16*n^2 = 16*A000290(n). - Omar E. Pol, Dec 11 2008

a(n) = 8*A001105(n) = 4*A016742(n) = 2*A139098(n). - Omar E. Pol, Dec 13 2008

a(n) = a(n-1) + 16*(2*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 20 2010

From Amiram Eldar, Jan 25 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/96.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/192.

Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/4)/(Pi/4).

Product_{n>=1} (1 - 1/a(n)) = sin(Pi/4)/(Pi/4) = 2*sqrt(2)/Pi (A112628). (End)

From Elmo R. Oliveira, Nov 30 2024: (Start)

G.f.: 16*x*(1 + x)/(1-x)^3.

E.g.f.: 16*x*(1 + x)*exp(x).

a(n) = n*A008598(n) = A195146(2*n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)