A182834 -id:A182834 - cp's OEIS frontend

A007590 a(n) = floor(n^2/2).

0, 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, 72, 84, 98, 112, 128, 144, 162, 180, 200, 220, 242, 264, 288, 312, 338, 364, 392, 420, 450, 480, 512, 544, 578, 612, 648, 684, 722, 760, 800, 840, 882, 924, 968, 1012, 1058, 1104, 1152, 1200, 1250, 1300, 1352, 1404

Offset: 0

N. J. A. Sloane, R. K. Guy

Arithmetic mean of a pair of successive triangular numbers. - Amarnath Murthy, Jul 24 2005
Maximum sum of absolute differences of cyclically adjacent elements in a permutation of (1..n). For example, with n = 9, permutation (1,9,2,8,3,7,4,6,5) has adjacent differences (8,7,6,5,4,3,2,1,4) with maximal sum a(9) = 40. - Joshua Zucker, Dec 15 2005
a(n) = maximum number of non-overlapping 1 X 2 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 tool, see links. - Dmitry Kamenetsky, Aug 03 2009 [This is easily provable - David W. Wilson, Jan 25 2014]
Number of strictly increasing arrangements of 3 nonzero numbers in -(n+1)..(n+1) with sum zero. For example, a(2) = 2 has two solutions: (-3, 1, 2) and (-2, -1, 3) each add to zero. - Michael Somos, Apr 11 2011
For n >= 4 is a(n) the minimal value v such that v = Sum_{i in S1} i = Product_{j in S2} j with disjoint union of S1, S2 = {1, 2, ..., n+1}. Example: a(4) = 8 = 3+5 = 1*2*4. - Claudio Meller, May 27 2012
Sum_{n > 1} 1/a(n) = (zeta(2) + 1)/2. - Enrique Pérez Herrero, Jun 19 2013
Apart from the initial term this is the elliptic troublemaker sequence R_n(2,4) 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
Maximum sum of displacements of elements in a permutation of (1..n). For example, with n = 9, permutation (5,6,7,8,9,1,2,3,4) has displacements (4,4,4,4,4,5,5,5,5) with maximal sum a(9) = 40. - David W. Wilson, Jan 25 2014
A245575(a(n)) mod 2 = 1, or for n > 0, where odd terms occur in A245575. - Reinhard Zumkeller, Aug 05 2014
Also the matching number of the n X n king, rook, and rook complement graphs. - Eric W. Weisstein, Jun 20 and Sep 14 2017
For n > 1, also the vertex count of the n X n white bishop graph. - Eric W. Weisstein, Jun 27 2017
This is also the number of distinct ways n^2 can be represented as the sum of two positive integers. - William Boyles, Jan 15 2018
Also the crossing number of the complete bipartite graph K_{4,n+1}. - Eric W. Weisstein, Sep 11 2018
The sequence can be obtained from A033429 by deleting the last digit of each term. - Bruno Berselli, Sep 11 2019
Starting at n=2, the number of facets of the n-dimensional Kunz cone C_(n+1). - Emily O'Sullivan, Jul 08 2023

			a(3) = 4 because 3^2/2 = 9/2 = 4.5 and floor(4.5) = 4.
a(4) = 8 because 4^2/2 = 16/2 = 8.
a(5) = 12 because 5^2/2 = 25/2 = 12.5 and floor(12.5) = 12.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
Emily O'Sullivan, Understanding the face structure of the Kunz cone, Master's thesis, San Diego State Univ., 2023.
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, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Vertex Count.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column 3 of triangle A094953.
For n > 2: a(n) = sum of (n-1)-th row in triangle A101037.
Cf. A000212, A000290, A056827, A118013, A118015.
A080476 is essentially the same sequence.
Cf. A000982.
Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), 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)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A182834 (complement), A245575.
First differences: A052928(n+1), is first differences of A212964; partial sums: A212964(n+1), is partial sums of A052928. - Guenther Schrack, Dec 10 2017
Cf. A033429 (5*n^2).
Cf. A056834, A130519, A056838, A056865.

a007550 = flip div 2 . (^ 2)  -- Reinhard Zumkeller, Aug 05 2014

a007590 = 0 : 0 : 0 : [ a1 + a2 - a3 + 2 | (a1, a2, a3) <- zip3 (tail (tail a007590)) (tail a007590) a007590 ] -- Luc Duponcheel, Sep 30 2020

[Floor(n^2/2): n in [0..53]]; // Bruno Berselli, Mar 28 2011

[Binomial(n,2)+Floor(n/2): n in [0..60]]; // Bruno Berselli, Jun 08 2017

A007590:=n->floor(n^2/2); seq(A007590(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Floor[Range[0, 53]^2/2] (* Alonso del Arte, Aug 07 2013 *)
Table[Binomial[n, 2] + Floor[n/2], {n, 0, 60}] (* Bruno Berselli, Jun 08 2017 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 2, 4, 8}, 20] (* Eric W. Weisstein, Sep 14 2017 *)
CoefficientList[Series[-2 x/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 14 2017 *)
Table[Floor[n^2/2], {n, 0, 20}] (* Eric W. Weisstein, Sep 11 2018 *)

PARI
```
{a(n) = n^2 \ 2}
```

{a(n) = local(v, c, m); m = n+1; forvec( v = vector( 3, i, [-m, m]), if( 0==prod( k=1, 3, v[k]), next); if( 0==sum( k=1, 3, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

first(n) = Vec(2*x^2/((1+x)*(1-x)^3) + O(x^n), -n); \\ Iain Fox, Dec 11 2017

def A007590(n): return n**2//2 # Chai Wah Wu, Jun 07 2022

a(n) = a(n-1) + a(n-2) - a(n-3) + 2 = 2*A002620(n) = A000217(n+1) + A004526(n). - Henry Bottomley, Mar 08 2000
a(n+1) = Sum_{k=1..n} (k + (k mod 2)). Therefore a(n) = Sum_{k=1..n} 2*floor(k/2). - William A. Tedeschi, Mar 19 2008
From R. J. Mathar, Nov 22 2008: (Start)
G.f.: 2*x^2/((1+x)*(1-x)^3).
a(n+1) - a(n) = A052928(n+1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - R. H. Hardin, Mar 28 2011
a(n) = (2*n^2 + (-1)^n - 1)/4. - Bruno Berselli, Mar 28 2011
a(n) = ceiling((n^2-1)/2) = binomial(n+1, 2) - ceiling(n/2). - Wesley Ivan Hurt, Mar 08 2014, Jun 14 2013
a(n+1) = A014105(n) - A032528(n). - Richard R. Forberg, Aug 07 2013
a(n) = binomial(n,2) + floor(n/2). - Bruno Berselli, Jun 08 2017
a(n) = A099392(n+1) - 1. - Guenther Schrack, Dec 10 2017
E.g.f.: (x*(x + 1)*cosh(x) + (x^2 + x - 1)*sinh(x))/2. - Stefano Spezia, May 06 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=2} 1/a(n) = Pi^2/12 + 1/2.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 - 1/2. (End)

Edited by Charles R Greathouse IV, Apr 20 2010

A078633 Smallest number of sticks of length 1 needed to construct n squares with sides of length 1.

Original entry on oeis.org

4, 7, 10, 12, 15, 17, 20, 22, 24, 27, 29, 31, 34, 36, 38, 40, 43, 45, 47, 49, 52, 54, 56, 58, 60, 63, 65, 67, 69, 71, 74, 76, 78, 80, 82, 84, 87, 89, 91, 93, 95, 97, 100, 102, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 125, 127, 130, 132, 134, 136, 138, 140, 142

Offset: 1

Mambetov Timur and Takenov Nurdin (timur_teufel(AT)mail.ru), Dec 12 2002

A182834(a(n)) mod 2 = 0, or, where even terms occur in A182834. - Reinhard Zumkeller, Aug 05 2014

			a(2)=7 because we have following construction:
   _ _
  |_|_|

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralph H. Buchholz and Warwick De Launey, The square, the triangle and the hexagon, 1996.
Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2019.
Douglas A. Torrance, Enumeration of planar Tangles, Proc. Math. Sci. 130 (50) (2020).

Cf. A027434, A027709, A135708, A182834.

a078633 n = 2 * n + ceiling (2 * sqrt (fromIntegral n))
-- Reinhard Zumkeller, Aug 05 2014

Table[2n+Ceiling[2Sqrt[n]],{n,70}] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = 2*n + ceil(2*sqrt(n)); \\ Michel Marcus, Mar 26 2018

from math import isqrt
def A078633(n): return (m:=n<<1)+1+isqrt((m<<1)-1) # Chai Wah Wu, Jul 28 2022

a(n) = 2*n + ceiling(2*sqrt(n)) = 2*n + A027434(n).

a(n) = (4*n + A027709(n))/2. - Tanya Khovanova, Mar 07 2008

A245575 Number of ways of writing n as the sum of two quarter-squares (cf. A002620).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 4, 2, 2, 4, 2, 5, 0, 4, 4, 4, 0, 3, 4, 4, 2, 2, 4, 2, 4, 5, 0, 4, 0, 6, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 0, 4, 2, 5, 4, 2, 2, 2, 4, 4, 2, 6, 0, 3, 4, 4, 0, 2, 6, 4, 2, 4, 2, 2, 0, 7, 4, 4, 0, 6, 0, 4, 2, 2, 6, 2, 2, 5, 4

Offset: 0

Reinhard Zumkeller, Aug 04 2014

nonn

a(n) is also the number of times n appears in the triangle A338796, or equivalently, the number of positive integer solutions of the equation A338796(x, y) = n for y <= x. - Stefano Spezia, Mar 03 2022

			a(10) = #{9+1, 6+4, 4+6, 1+9} = 4;
a(11) = #{9+2, 2+9} = 2;
a(12) = #{12+0, 6+6, 0+12} = 3;
a(13) = #{12+1, 9+4, 4+9, 1+12} = 4;
a(14) = #{6+1, 1+6} = 2;
a(15) = #{9+6, 6+9} = 2;
a(16) = #{16+0, 12+4, 4+12, 0+16} = 4;
a(17) = #{16+1, 1+16} = 2;
a(18) = #{16+2, 12+6, 9+9, 6+12, 2+16} = 5;
a(19) = #{} = 0;
a(20) = #{20+0, 16+4, 4+16, 0+20} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A002620, A007550, A182834, A240025, A240952, A245585, A338796.

a245575 n = a245575_list !! n
a245575_list = f 0 [] $ tail a002620_list where
   f u vs ws'@(w:ws)
     | u < w     = (sum $ map (a240025 . (u -)) vs) : f (u + 1) vs ws'
     | otherwise = f u (w : vs) ws

qsQ[n_] := qsQ[n] = With[{s = Sqrt[n]}, Which[IntegerQ[s], True, n == Floor[s] (Floor[s]+1), True, True, False]]; a[n_] := Count[Range[0, n], k_ /; qsQ[k] && qsQ[n-k]]; Array[a, 100, 0] (* Jean-François Alcover, May 08 2017 *) (* or *)
u[{x_,y_}] := 2-Boole[x==y]; a[n_] := Total[u /@ IntegerPartitions[n, {2}, Floor[Range[1 + 2 Sqrt@ n]^2/4]]]; Array[a, 100, 0] (* Giovanni Resta, May 08 2017 *)

a(A182834(n)) mod 2 = 0; a(A007550(n)) mod 2 = 1;
a(A240952(n)) = n and a(A240952(m)) <> n for m < a(n);
a(A245585(n)) = 0.

A349080 Numbers k for which there exists only one integer m with 1 <= m <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.

Original entry on oeis.org

1, 2, 4, 12, 18, 20, 24, 28, 34, 36, 40, 44, 52, 56, 60, 62, 64, 68, 76, 80, 84, 88, 92, 98, 100, 104, 108, 112, 116, 120, 124, 132, 136, 140, 142, 144, 148, 152, 156, 164, 168, 172, 176, 180, 184, 188, 192, 194, 196, 204, 208, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 254, 256

Offset: 1

Bernard Schott, Nov 20 2021

nonn

This sequence is the union of {1} and of three infinite and disjoint subsequences.
-> Numbers k divisible by 4 but not of the form 8q^2 or 8q(q+1) = {4, 12, 20, 24, 28, ...} (see A182834). For these numbers, the corresponding unique m = k/2 (see example for k = 4).
-> Even numbers k not divisible by 4 and of the form k = 2*A055792 = 2*q^2, q>1 in A001541 = {18, 578, ...}. For these numbers, the corresponding unique m = k/2 - 2 = q^2-2 (see example for k = 18)
-> Even numbers k not divisible by 4, that are in A060626 but not of the form k=2q^2-4 with q>1 in A001541 = {2, 34, 62, 98, 142, 194, ...} (A349496). For these numbers, the corresponding unique m = k/2 + 1 (see example for k = 2).
See A348692 for further information.

			For k = 2, 2$ / 2! = 1^2, hence 2 is a term.
For k = 4, 4$ /1! = 288, 4$ / 3! = 48, 4$ / 4! = 12 but for m = 2, 4$ / 2! = 12^2, hence 4 is a term.
For k = 18 and m = 7, we have 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2 and there is no other solution m, hence 18 is a term.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.

Cf. A000178, A001541, A060626, A348692, A182834, A349496.

Subsequence of A349079.

q[n_] := Count[BarnesG[n + 2]/Range[n]!, ?(IntegerQ@Sqrt[#] &)] == 1; Select[Range[100], q] (* _Amiram Eldar, Nov 20 2021 *)

sf(n) = prod(k=2, n, k!); \\ A000178
isok(m) = my(s=sf(m)); #select(issquare, vector(m, k, s/k!), 1) == 1; \\ Michel Marcus, Nov 20 2021

cp's OEIS Frontend

A007590 a(n) = floor(n^2/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

A078633 Smallest number of sticks of length 1 needed to construct n squares with sides of length 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A245575 Number of ways of writing n as the sum of two quarter-squares (cf. A002620).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A349080 Numbers k for which there exists only one integer m with 1 <= m <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A349080 Numbers k for which there exists only one integer m with 1 <= m <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.