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a(n) = 20*n^2.
11

%I #86 Feb 18 2024 08:29:21

%S 0,20,80,180,320,500,720,980,1280,1620,2000,2420,2880,3380,3920,4500,

%T 5120,5780,6480,7220,8000,8820,9680,10580,11520,12500,13520,14580,

%U 15680,16820,18000,19220,20480,21780,23120,24500,25920,27380,28880,30420,32000

%N a(n) = 20*n^2.

%C 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.

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A195322/b195322.txt">Table of n, a(n) for n = 0..10000</a>

%H Léo Sauvé, <a href="https://cms.math.ca/crux/backfile/Crux_v1n09_Nov.pdf">Problem 53</a>, Crux Mathematicorum, page 88, Vol.1, Nov. 75.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).

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

%F 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

%F a(n) = A010014(n) - A005899(n) for n>0. - _R. J. Cano_, Sep 29 2015

%e From _Muniru A Asiru_, Feb 01 2018: (Start)

%e n=0, a(0) = 20*0^2 = 0.

%e n=1, a(1) = 20*1^2 = 20.

%e n=1, a(2) = 20*2^2 = 80.

%e n=1, a(3) = 20*3^2 = 180.

%e n=1, a(4) = 20*4^2 = 320.

%e ...

%e (End)

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

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

%o (Magma) [20*n^2: n in [0..40]]; // _Vincenzo Librandi_, Sep 20 2011

%o (PARI) a(n) = 20*n^2 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (GAP) List([0..10^3],n->20*n^2); # _Muniru A Asiru_, Feb 01 2018

%Y Bisection of A195148.

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

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Sep 16 2011