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%I #103 Feb 03 2021 09:01:21
%S 0,5,20,45,80,125,180,245,320,405,500,605,720,845,980,1125,1280,1445,
%T 1620,1805,2000,2205,2420,2645,2880,3125,3380,3645,3920,4205,4500,
%U 4805,5120,5445,5780,6125,6480,6845,7220,7605,8000,8405,8820,9245,9680,10125,10580,11045,11520,12005,12500
%N a(n) = 5*n^2.
%C Number of edges of the complete bipartite graph of order 6n, K_n,5n. - _Roberto E. Martinez II_, Jan 07 2002
%C Number of edges of the complete tripartite graph of order 4n, K_n,n,2n. - _Roberto E. Martinez II_, Jan 07 2002
%C a(n+1)-a(n) : 5, 15, 25, 35, 45, ... (see A017329). - _Philippe Deléham_, Dec 08 2011
%C From _Larry J Zimmermann_, Feb 21 2013: (Start)
%C The sum of the areas of 2 squares that equals the area of a rectangle with whole number sides using the formula x^2 + y^2 = (x+y+sqrt(2*x*y))(x+y-sqrt(2*x*y)), where the substitution y=2*x obtains the whole number sides of the rectangle. So x^2+(2*x)^2=5x(x).
%C x squares sum rectangle (l,w) area
%C 1 1,4 5 5,1 5
%C 2 4,16 20 10,2 20 (End)
%H G. C. Greubel, <a href="/A033429/b033429.txt">Table of n, a(n) for n = 0..5000</a>
%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 5*A000290(n). - _Omar E. Pol_, Dec 11 2008
%F From _Bruno Berselli_, Feb 11 2011: (Start)
%F G.f.: 5*x*(1+x)/(1-x)^3.
%F a(n) = 4*A000217(n) + A000567(n). (End)
%F a(n) = a(n-1)+5*(2*n-1) (with a(0)=0). - _Vincenzo Librandi_, Nov 17 2010
%F a(n) = A131242(10*n+4). - _Philippe Deléham_, Mar 27 2013
%F a(n) = a(n-1) + 10*n - 5, with a(0)=0. - _Jean-Bernard François_, Oct 04 2013
%F a(n) = A001105(n) + A033428(n). - _Altug Alkan_, Sep 28 2015
%F E.g.f.: 5*x*(x+1)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F a(n) = Sum_{i = 2..6} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - _Bruno Berselli_, Jul 04 2018
%F From _Amiram Eldar_, Feb 03 2021: (Start)
%F Sum_{n>=1} 1/a(n) = Pi^2/30.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/60.
%F Product_{n>=1} (1 + 1/a(n)) = sqrt(5)*sinh(Pi/sqrt(5))/Pi.
%F Product_{n>=1} (1 - 1/a(n)) = sqrt(5)*sin(Pi/sqrt(5))/Pi. (End)
%t 5*Range[50]^2 (* _Alonso del Arte_, May 23 2012 *)
%o (PARI) a(n)=5*n^2
%Y Central column of A055096.
%Y Cf. A000290.
%Y Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A022268, A049452, A186030, A135703, A152734, A139273.
%Y Cf. A185019.
%Y Cf. A001105, A033428.
%Y Similar sequences are listed in A316466.
%K nonn,easy
%O 0,2
%A _Jeff Burch_
%E Better description from _N. J. A. Sloane_, May 15 1998
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