A033429 a(n) = 5*n^2.

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

Offset: 0

Author

Jeff Burch

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Central column of A055096.

Cf. A000290.

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.

Cf. A185019.

Cf. A001105, A033428.

Similar sequences are listed in A316466.

Programs

Mathematica

5*Range[50]^2 (* Alonso del Arte, May 23 2012 *)

PARI

a(n)=5*n^2

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)

Extensions

Better description from N. J. A. Sloane, May 15 1998