A118015 a(n) = floor(n^2/5).

0, 0, 0, 1, 3, 5, 7, 9, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 64, 72, 80, 88, 96, 105, 115, 125, 135, 145, 156, 168, 180, 192, 204, 217, 231, 245, 259, 273, 288, 304, 320, 336, 352, 369, 387, 405, 423, 441, 460, 480, 500, 520, 540, 561, 583, 605, 627, 649, 672

Offset: 0

Author

Reinhard Zumkeller, Apr 10 2006

Keywords

• nonn
• easy

Comments

It seems that for n >= 5, a(n) is the maximum number of non-overlapping 1 X 5 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's program. - Dmitry Kamenetsky, Aug 03 2009

Ismailescu & Lee prove that for n > 6, a(n) is composite. - Charles R Greathouse IV, Jan 10 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 1.
• R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Crossrefs

Cf. A000212, A000290, A007590, A002620, A056827, A118013, A279169.

Cf. A056834, A130519, A056838, A056865.

Programs

Magma

[ n^2 div 5: n in [0..58] ]; // Klaus Brockhaus, Nov 18 2008

Mathematica

Table[Floor[n^2/5], {n, 0, 60}] (* Bruno Berselli, Dec 12 2016 *)

PARI

a(n)=n^2\5 \\ Charles R Greathouse IV, Sep 24 2015

Python

[int(n**2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

Sage

[floor(n^2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

Formula

G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^3). - Klaus Brockhaus, Nov 18 2008

a(n) = A008732(n-4) + A008732(n-3). - R. J. Mathar, Nov 22 2008

a(5*m+r) = m*(5*m + 2*r) + a(r), with m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*(5*4 + 2*3) + a(3) = 104 + 1 = 105. - Bruno Berselli, Dec 12 2016

Sum_{n>=3} 1/a(n) = 25/16 + Pi^2/30 + sqrt(5-2*sqrt(5))*Pi/4. - Amiram Eldar, Aug 13 2022

Extensions

Edited by Charles R Greathouse IV, Apr 20 2010