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%I #83 Sep 08 2022 08:44:59
%S 0,4,22,70,170,350,644,1092,1740,2640,3850,5434,7462,10010,13160,
%T 17000,21624,27132,33630,41230,50050,60214,71852,85100,100100,117000,
%U 135954,157122,180670,206770,235600,267344,302192,340340,381990,427350,476634
%N Number of nonsquare rectangles on an n X n board.
%C Partial sums of A045991 (n^3-n^2). - _Jeremy Gardiner_, Jun 30 2013
%H Vincenzo Librandi, <a href="/A052149/b052149.txt">Table of n, a(n) for n = 1..1000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=6">Sum square difference: Problem 6</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n*(n-1)*(n+1)*(3*n+2)/12.
%F G.f.: 2*x^2*(2+x)/(1-5*x+10*x^2-10*x^3+5*x^4-x^5). - _Colin Barker_, Jan 04 2012
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Vincenzo Librandi_, Apr 28 2012
%F a(n) = A033487(n-1) - A007290(n+1) starting at n=1. - _J. M. Bergot_, Jun 04 2012
%F a(n) = Sum_{k=1..n} (k-1)*k^2. - _Michel Marcus_, Nov 09 2012
%F a(n) = A000537(n) - A000330(n) = 2*A000914(n-1). - _Luciano Ancora_, Mar 16 2015
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 81*log(3)/5 - 9*sqrt(3)*Pi/5 - 192/25.
%F Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 - 48*log(2)/5 - 318/25. (End)
%e a(10) = 10 * 9 * 11 * 32 / 12 = 2640.
%e a(5) = 170 and the sum from 1 to 5 is 15, giving 1*(15-1)=14, 2*(15-2)=26, 2*(15-3)=36, 4*(15-4)=44 and 5*(15-5)=50; adding 14+26+36+44+50=170. Do the same for each n and get a(n). - _J. M. Bergot_, Oct 31 2014
%p a:=n->sum(j^3-j^2, j=0..n): seq(a(n), n=1..37); # _Zerinvary Lajos_, May 08 2008
%t CoefficientList[Series[2*x*(2+x)/(1-5*x+10*x^2-10*x^3+ 5*x^4-x^5), {x,0,50}], x] (* _Vincenzo Librandi_, Apr 28 2012 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,4,22,70,170},40] (* _Harvey P. Dale_, Jul 30 2019 *)
%o (Magma) I:=[0, 4, 22, 70, 170]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // _Vincenzo Librandi_, Apr 28 2012
%o (PARI) a(n) = sum(k=1,n,(k-1)*k^2) \\ _Michel Marcus_, Nov 09 2012
%Y Cf. A035291, A045991, A033487, A007290, A000537, A000330, A000914.
%K nonn,easy
%O 1,2
%A Ronald Arms (ron.arms(AT)stanfordalumni.org), Jan 23 2000
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