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A033429 a(n) = 5*n^2. 40
0, 5, 20, 45, 80, 125, 180, 245, 320, 405, 500, 605, 720, 845, 980, 1125, 1280, 1445, 1620, 1805, 2000, 2205, 2420, 2645, 2880, 3125, 3380, 3645, 3920, 4205, 4500, 4805, 5120, 5445, 5780, 6125, 6480, 6845, 7220, 7605, 8000, 8405, 8820, 9245, 9680, 10125, 10580, 11045, 11520, 12005, 12500 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of edges of the complete bipartite graph of order 6n, K_n,5n. - Roberto E. Martinez II, Jan 07 2002
Number of edges of the complete tripartite graph of order 4n, K_n,n,2n. - Roberto E. Martinez II, Jan 07 2002
a(n+1)-a(n) : 5, 15, 25, 35, 45, ... (see A017329). - Philippe Deléham, Dec 08 2011
From Larry J Zimmermann, Feb 21 2013: (Start)
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).
x squares sum rectangle (l,w) area
1 1,4 5 5,1 5
2 4,16 20 10,2 20 (End)
LINKS
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
FORMULA
a(n) = 5*A000290(n). - Omar E. Pol, Dec 11 2008
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: 5*x*(1+x)/(1-x)^3.
a(n) = 4*A000217(n) + A000567(n). (End)
a(n) = a(n-1)+5*(2*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
a(n) = A131242(10*n+4). - Philippe Deléham, Mar 27 2013
a(n) = a(n-1) + 10*n - 5, with a(0)=0. - Jean-Bernard François, Oct 04 2013
a(n) = A001105(n) + A033428(n). - Altug Alkan, Sep 28 2015
E.g.f.: 5*x*(x+1)*exp(x). - G. C. Greubel, Jul 17 2017
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
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/30.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/60.
Product_{n>=1} (1 + 1/a(n)) = sqrt(5)*sinh(Pi/sqrt(5))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(5)*sin(Pi/sqrt(5))/Pi. (End)
MATHEMATICA
5*Range[50]^2 (* Alonso del Arte, May 23 2012 *)
PROG
(PARI) a(n)=5*n^2
CROSSREFS
Central column of A055096.
Cf. A000290.
Cf. A185019.
Similar sequences are listed in A316466.
Sequence in context: A228168 A178977 A061188 * A168011 A160749 A147002
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description from N. J. A. Sloane, May 15 1998
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)