A195322 a(n) = 20*n^2.

0, 20, 80, 180, 320, 500, 720, 980, 1280, 1620, 2000, 2420, 2880, 3380, 3920, 4500, 5120, 5780, 6480, 7220, 8000, 8820, 9680, 10580, 11520, 12500, 13520, 14580, 15680, 16820, 18000, 19220, 20480, 21780, 23120, 24500, 25920, 27380, 28880, 30420, 32000, 33620, 35280

Offset: 0

Author

Omar E. Pol, Sep 16 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 20, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semiaxis opposite to A195317 in the same spiral.

a(n) is the sum of all the integers less than 10*n which are not multiple of 2 or 5. a(2) = (1 + 3 + 7 + 9) + (11 + 13 + 17 + 19) = 20 + 60 = 80 = 20 * 2^2. (Link Crux Mathematicorum). - Bernard Schott, May 15 2017

Number of terms less than 10^k (k=0, 1, 2, ...): 1, 1, 3, 8, 23, 71, 224, 708, 2237, 7072, 22361, 70711, ... - Muniru A Asiru, Feb 01 2018

Examples

			From _Muniru A Asiru_, Feb 01 2018: (Start)
n=0, a(0) = 20*0^2 = 0.
n=1, a(1) = 20*1^2 = 20.
n=1, a(2) = 20*2^2 = 80.
n=1, a(3) = 20*3^2 = 180.
n=1, a(4) = 20*4^2 = 320.
...
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Léo Sauvé, Problem 53, Crux Mathematicorum, Vol. 1, Nov. 1975, page 88.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A195148.

Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195323.

Cf. A000290, A001105, A005899, A010014, A016742, A033429, A195162, A195317.

Cf. A008602.

Programs

GAP

List([0..10^3],n->20*n^2); # Muniru A Asiru, Feb 01 2018

Magma

[20*n^2: n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

Maple

a := n -> 20*n^2; seq(a(n), n=0..10^3); # Muniru A Asiru, Feb 01 2018

Mathematica

20 Range[0, 40]^2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 20, 80}, 50] (* Harvey P. Dale, Jan 18 2013 *)

PARI

a(n) = 20*n^2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).

a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013

a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015

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

G.f.: 20*x*(1 + x)/(1-x)^3.

E.g.f.: 20*x*(1 + x)*exp(x).

a(n) = n*A008602(n) = A195148(2*n). (End)