A279169 -id:A279169 - cp's OEIS frontend

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

Reinhard Zumkeller, Apr 10 2006

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

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).

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

Cf. A056834, A130519, A056838, A056865.

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

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

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

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

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

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

Edited by Charles R Greathouse IV, Apr 20 2010

A033437 Number of edges in 5-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 14, 19, 25, 32, 40, 48, 57, 67, 78, 90, 102, 115, 129, 144, 160, 176, 193, 211, 230, 250, 270, 291, 313, 336, 360, 384, 409, 435, 462, 490, 518, 547, 577, 608, 640, 672, 705, 739, 774, 810, 846, 883, 921, 960, 1000, 1040, 1081, 1123, 1166, 1210, 1254

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,5) (also sequence R_n(4,5)) in the notation of Stange (see Table 1, p. 16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 12 2013

R. L. Graham et al., eds., Handbook of Combinatorics, Vol. 2, p. 1234.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Turán Graph
Wikipedia, Turán graph
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033438, A033439, A033440, A033441, A033442, A033443, A033444. - Reinhard Zumkeller, Nov 30 2009
Elliptic troublemaker sequences: A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A279169.

[2*n^2 div 5: n in [0..60]]; // Vincenzo Librandi, Apr 20 2015

Mathematica
```
Table[Floor[2n^2/5],{n,0,60}]
```

a(n)=2*n^2\5 \\ Charles R Greathouse IV, Apr 20 2015

G.f.: (x^5+x^4+x^3+x^2)/((1-x^5)*(1-x)^2).
a(n) = Sum_{k=0..n} A011558(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = floor( 2n^2/5 ). - Wesley Ivan Hurt, Jun 20 2013
a(n) = Sum_{i=1..n} floor(4*i/5). - Wesley Ivan Hurt, Sep 12 2017

A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

Original entry on oeis.org

1, 2, 5, 9, 15, 21, 29, 38, 48, 60, 72, 86, 101, 117, 135, 153, 173, 194, 216, 240, 264, 290, 317, 345, 375, 405, 437, 470, 504, 540, 576, 614, 653, 693, 735, 777, 821, 866, 912, 960, 1008, 1058, 1109, 1161, 1215, 1269, 1325, 1382, 1440, 1500, 1560, 1622, 1685, 1749, 1815, 1881, 1949, 2018, 2088, 2160, 2232, 2306, 2381

Offset: 1

Clark Kimberling, Jan 16 2011

Apart from the initial term this is the elliptic troublemaker sequence R_n(2,5) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences see the cross references below. - Peter Bala, Aug 08 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT]
Wikipedia, Maximum number of deciamonds in a hexagon.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 1, -2, 1).

Cf. A183532, A279169.

Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A184535 (= R_n(2,5) = R_n(3,5)).

Concatenation([1], List([2..10^3], n->Int(3/5 * n^2))); # Muniru A Asiru, Feb 04 2018

1,seq(floor(3/5*n^2), n=2..10^3); # Muniru A Asiru, Feb 04 2018

p[n_] := FractionalPart[(n^3 + 5)^(1/3)]; q[n_] := Floor[1/p[n]]; Table[q[n], {n, 1, 120}]
Join[{1},LinearRecurrence[{2, -1, 0, 0, 1, -2, 1},{2, 5, 9, 15, 21, 29, 38},62]] (* Ray Chandler, Aug 31 2015 *)

a(n) = if(n==1, 1, 3*n^2\5); \\ Altug Alkan, Mar 03 2018

def A184535(n): return 3*n**2//5 if n>1 else 1 # Chai Wah Wu, Aug 04 2025

a(n) = floor(1/{(5+n^3)^(1/3)}), where {}=fractional part.
a(n)= +2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7), for n>8, with g.f. 1-x^2*(1+x)*(2*x^2-x+2)/ ((x^4+x^3+x^2+x+1) *(x-1)^3), so a(n) is (3n^2-2)/5 plus a fifth of A164116 for n>1. [Bruno Berselli, Jan 30 2011. See the following Bala's comment.]
From Peter Bala, Aug 08 2013: (Start)
a(n) = floor(3/5*n^2) for n >= 2.
The sequence b(n) := floor(3/5*n^2) - 3/5*n^2, n >= 1, is periodic with period [-3/5, -2/5, -2/5, -3/5, 0] of length 5. The generating function and recurrence equation given above easily follow from these observations.
The sequence c(n) := 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ) is also periodic with period 5, and calculation shows it has the same period as the sequence b(n). Thus b(n) = c(n), yielding the alternative formula a(n) = 3/5*n^2 + 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ), which is one of the formulas for the elliptic troublemaker sequence R_n(2,5) given in Stange (see Section 7, equation (21)). (End)

Better name from Peter Bala, Aug 08 2013

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

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A033437 Number of edges in 5-partite Turán graph of order n.

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A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

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