A007590 -id:A007590 - cp's OEIS frontend

A099392 a(n) = floor((n^2 - 2*n + 3)/2).

1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405

Offset: 1

Ralf Stephan following a suggestion from Luke Pebody, Oct 20 2004

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Differs from A085913 at n = 61. Apart from leading term, identical to A080827.
Cf. A000217, A001844, A002522, A007494, A007590, A058331 (bisections).
From Guenther Schrack, Apr 17 2018: (Start)
First differences: A052928.
Partial sums: A212964(n) + n for n > 0.
Also A058331 and A001844 interleaved. (End)
Cf. A014601, A033683, A047838, A074148, A109613, A183575, A290743.

Array[Floor[(#^2 - 2 # + 3)/2] &, 54] (* or *)
Rest@ CoefficientList[Series[x (-1 + x - x^2 - x^3)/((1 + x) (x - 1)^3), {x, 0, 54}], x] (* Michael De Vlieger, Apr 21 2018 *)

a(n)=(n^2+3)\2-n \\ Charles R Greathouse IV, Aug 01 2013

a(n) = ceiling(n^2/2)-n+1. - Paul Barry, Jul 16 2006; index shifted by R. J. Mathar, Jul 29 2007
a(n) = ceiling(A002522(n-1)/2). - Branko Curgus, Sep 02 2007
From R. J. Mathar, Feb 20 2011: (Start)
G.f.: x *( -1+x-x^2-x^3 ) / ( (1+x)*(x-1)^3 ).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n+1) = (3 + 2*n^2 + (-1)^n)/4. (End)
a(n) = A007590(n-1) + 1 for n >= 2. - Richard R. Forberg, Aug 01 2013
a(n) = A000217(n) - A007494(n-1). - Bui Quang Tuan, Mar 27 2015
From Guenther Schrack, Apr 17 2018: (Start)
a(n) = (2*n^2 - 4*n + 5 -(-1)^n)/4.
a(n+2) = a(n) + 2*n for n > 0.
a(n) = 2*A033683(n-1) - 1 for n > 0.
a(n) = A047838(n-1) + 2 for n > 2.
a(n) = A074148(n-1) - n + 2 for n > 1.
a(n) = A183575(n-3) + 3 for n > 3.
a(n) = 2*A290743(n-1) - 3 for n > 0.
a(n) = 2*A290743(n-2) + A109613(n-5) for n > 4.
a(n) = A074148(n) - A014601(n-1) for n > 0. (End)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 16 2022
E.g.f.: ((2 - x + x^2)*cosh(x) + (3 - x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 28 2024

A173196 Partial sums of A002620.

Original entry on oeis.org

0, 0, 1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

Jonathan Vos Post, Feb 12 2010

Essentially a duplicate of A002623: 0, 0, followed by A002623.
The only primes in this sequence are 3, 7, and 13: for n > 2 both a(2*n+1) = n*(n+1)*(4*n+5)/6 and a(2*n) = n*(n+1)*(4*n-1)/6 are composite. - Bruno Berselli, Jan 19 2011
a(n-1) is the number of integer-sided scalene triangles with largest side <= n, including degenerate (i.e., collinear) triangles. a(n-2) is the number of non-degenerate integer-sided scalene triangles. - Alexander Evnin, Oct 12 2010
Also n-th differences of square pyramidal numbers (A000330) and numbers of triangles in triangular matchstick arrangement of side n (A002717). - Konstantin P. Lakov, Apr 13 2018
Also the number of undirected bishop moves on a n X n chessboard, counted up to rotations and reflections of the board. - Hilko Koning, Aug 16 2025

			a(57) = 0 + 0 + 1 + 2 + 4 + 6 + 9 + 12 + 16 + 20 + 25 + 30 + 36 + 42 + 49 + 56 + 64 + 72 + 81 + 90 + 100 + 110 + 121 + 132 + 144 + 156 + 169 + 182 + 196 + 210 + 225 + 240 + 256 + 272 + 289 + 306 + 324 + 342 + 361 + 380 + 400 + 420 + 441 + 462 + 484 + 506 + 529 + 552 + 576 + 600 + 625 + 650 + 676 + 702 + 729 + 756 + 784 + 812 = 15834.

A. Yu. Evnin. Problem book on discrete mathematics. Moscow: Librokom, 2010; problem 787. (In Russian)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven.
M. Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Int. Seq. 14 (2011) # 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000212, A000330, A002620, A002623, A002717, A002984, A007590, A024206, A056827, A072280, A087811, A118013, A118015.

[Floor((2*n^3+3*n^2-2*n)/24): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

CoefficientList[Series[x^2/((1 - x)^3 (1 - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
Accumulate[Floor[Range[0,60]^2/4]] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,0,1,3,7},60] (* Harvey P. Dale, Feb 09 2020 *)
a[ n_] := Quotient[2 n^3 + 3 n^2 - 2 n, 24]; (* Michael Somos, Jan 14 2021 *)

G.f.: x^2 / ((1-x)^3 * (1-x^2)).
a(n) = (4*n^3 + 6*n^2 - 4*n - 3 + 3*(-1)^n)/48. - Bruno Berselli, Jan 19 2011
a(n) = A002623(n-2) for n >= 2. - Martin von Gagern, Dec 05 2014
a(n) = Sum_{i=0..n} A002620(i) = Sum_{i=0..n} floor(i/2)*ceiling(i/2) = Sum_{i=0..n} floor(i^2/4).
a(n) = round((2*n^3 + 3*n^2 - 2*n)/24) = round((4*n^3 + 6*n^2 - 4*n - 3)/48) = floor((2*n^3 + 3*n^2 - 2*n)/24) = ceiling((2*n^3 + 3*n^2 - 2*n - 3)/24). - Mircea Merca, Nov 23 2010
a(n) = a(n-2) + n*(n-1)/2, n > 1. - Mircea Merca, Nov 25 2010
a(n) = floor(n/2)*(floor(n/2)+1)*(8*ceiling(n/2) - 2*n - 1)/6. - Alexander Evnin, Oct 12 2010
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jan 14 2021
E.g.f.: (x*(3 + 9*x + 2*x^2)*cosh(x) - (3 - 3*x - 9*x^2 - 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A080827 Rounded up staircase on natural numbers.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405, 1459

Offset: 1

Paul Barry, Feb 28 2003

Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.
Partial sums of A131055. - Paul Barry, Jun 14 2008
The same sequence arises in the triangular array of integers >= 1 according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array and the second row of that subarray (with apex a(n-1)) contains just two numbers, one odd one even. The one with the same (odd) parity as a(n-1) is a(n). - David James Sycamore, Jul 29 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. C. F. de Winter, Using the Student's t-test with extremely small sample sizes, Practical Assessment, Research & Evaluation, 18(10), 2013.
Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
David James Sycamore, Triangular array.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Apart from leading term identical to A099392.

Cf. A000027, A000982, A007590, A131055.

List([1..10],n->Int(n^2/2)+1); # Muniru A Asiru, Aug 02 2018

[n*(n+1)/2-Floor((n-1)/2) : n in [1..60]]; // Vincenzo Librandi, Aug 05 2013

A080827:=n->(n^2+2-(1-(-1)^n)/2)/2: seq(A080827(n), n=1..100); # Wesley Ivan Hurt, Sep 08 2015

CoefficientList[Series[(1 + x - x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Ceiling[(Range[100]^2 + 1)/2] (* Paolo Xausa, Jul 19 2025 *)

a(n) = ceiling((n^2+1)/2).
a(1) = 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) + 2n. - Amarnath Murthy, May 07 2003
From Paul Barry, Apr 12 2008: (Start)
G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).
a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by R. J. Mathar, Jul 14 2013] (End)
From Wesley Ivan Hurt, Sep 08 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.
a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.
a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) - 1/2. - Amiram Eldar, Sep 15 2022
E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 27 2024

A188122 Table read by downward antidiagonals: T(n,k) is the number of strictly increasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 2, 3, 0, 4, 4, 8, 4, 0, 5, 8, 16, 16, 16, 0, 6, 12, 31, 42, 52, 42, 0, 7, 18, 51, 90, 137, 152, 137, 0, 8, 24, 80, 172, 308, 426, 484, 426, 0, 9, 32, 118, 296, 624, 1032, 1398, 1536, 1398, 0, 10, 40, 167, 482, 1154, 2216, 3528, 4622, 5064, 4622, 0, 11, 50

Offset: 1

R. H. Hardin, Mar 21 2011

Table starts
     0,    0,     0,     0,      0,      0,      0,      0,       0,       0,       0, ...
     1,    2,     3,     4,      5,      6,      7,      8,       9,      10,      11, ...
     0,    2,     4,     8,     12,     18,     24,     32,      40,      50,      60, ...
     3,    8,    16,    31,     51,     80,    118,    167,     227,     302,     390, ...
     4,   16,    42,    90,    172,    296,    482,    740,    1092,    1554,    2154, ...
    16,   52,   137,   308,    624,   1154,   1999,   3278,    5144,    7772,   11387, ...
    42,  152,   426,  1032,   2216,   4376,   8044,  13994,   23210,   37030,   57086, ...
   137,  484,  1398,  3528,   7970,  16547,  32035,  58595,  102113,  170844,  275878, ...
   426, 1536,  4622, 12124,  28660,  62222, 126122, 241250,  439514,  767656, 1292864, ...
  1398, 5064, 15594, 42262, 103599, 233880, 493267, 982016, 1861168, 3379972, 5913676, ...

			Some solutions for n=8, k=6:
  -11 -12 -11 -11 -12 -10 -11 -12 -12  -9 -10 -11 -11 -12 -12  -7
   -9  -9 -10  -8  -6  -8  -9 -11  -9  -5  -8 -10  -8 -10 -10  -6
   -5  -8  -4  -4  -4  -5  -8  -4  -8  -4  -5  -4  -7  -9  -5  -5
   -4  -3  -2  -2  -1  -3  -3  -2   1  -2  -3  -1   1   1  -2  -3
    2   5   2   1   2  -1   1  -1   4   1   4   3   2   2   3  -2
    6   8   4   6   4   8   7   7   5   2   5   4   6   8   5   5
    9   9  10   7   5   9  11  11   9   8   8   8   7   9   9   8
   12  10  11  11  12  10  12  12  10   9   9  11  10  11  12  10

R. H. Hardin, Table of n, a(n) for n = 1..550
Louis Ng, Magic counting with inside-out polytopes, Master's Thesis, San Francisco State University, 2018.

Row 3 is A007590.

A188333 T(n,k)=Number of nondecreasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 4, 10, 0, 4, 8, 20, 40, 0, 5, 12, 37, 86, 197, 0, 6, 18, 61, 166, 424, 980, 0, 7, 24, 94, 288, 828, 2128, 5142, 0, 8, 32, 136, 472, 1488, 4238, 11200, 27632, 0, 9, 40, 191, 726, 2519, 7836, 22563, 60372, 152191, 0, 10, 50, 257, 1076, 4050, 13694, 42593

Offset: 1

R. H. Hardin Mar 28 2011

Table starts
......0......0......0.......0.......0.......0.......0........0........0
......1......2......3.......4.......5.......6.......7........8........9
......2......4......8......12......18......24......32.......40.......50
.....10.....20.....37......61......94.....136.....191......257......338
.....40.....86....166.....288.....472.....726....1076.....1534.....2130
....197....424....828....1488....2519....4050....6252.....9314....13479
....980...2128...4238....7836...13694...22786...36454....56314....84496
...5142..11200..22563...42593...76251..130453..214784...341988...528926
..27632..60372.122986..236130..431488..755434.1274786..2082546..3306612
.152191.333254.684809.1333130.2477726.4423012.7621670.12729304.20676601

			Some solutions for n=8 k=6
.-8...-9..-12..-11..-12...-9...-9...-6..-11...-8..-12..-12..-12...-9..-11...-9
.-5...-7...-7...-7...-5...-6...-8...-6...-7...-7...-6..-12..-10...-6..-10...-7
.-1...-6....1...-4...-2...-3...-3...-6...-4...-4...-4...-1...-6...-2...-4...-1
.-1....1....1...-4...-2...-3....3...-5...-4....1...-4....2....4....1...-4...-1
.-1....1....2....3...-2....2....3...-5....5....1....1....4....4....1....4....1
..3....2....3....6....4....4....3....5....6....3....2....4....5....3....6....5
..3....9....5....8....8....7....4...11....6....4...11....4....6....5....9....6
.10....9....7....9...11....8....7...12....9...10...12...11....9....7...10....6

R. H. Hardin, Table of n, a(n) for n = 1..430

Row 3 is A007590(n+1)

A056865 a(n) = floor(n^2/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 108, 115, 122, 129, 136, 144, 152, 160, 168, 176, 184, 193, 202, 211, 220, 230, 240, 250, 260, 270, 280, 291, 302, 313, 324

Offset: 0

N. J. A. Sloane, Sep 02 2000

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A056864.

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A056834, A130519, A056838.

A056865 := proc(n)
    floor(n^2/10) ;
end proc:
seq(A056865(n),n=0..100) ; # R. J. Mathar, Mar 08 2016

Floor[Range[0,60]^2/10] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,1,2,3,4,6,8,10,12},60] (* Harvey P. Dale, Jun 29 2022 *)

a(n) = n^2\10; \\ Michel Marcus, Mar 08 2016

G.f.: -x^4*(1+x^4) / ( (1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^3 ). - R. J. Mathar, Mar 08 2016

A368434 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {0,1,...,n}.

Original entry on oeis.org

1, 2, 4, 2, 3, 8, 10, 4, 2, 4, 12, 18, 16, 8, 4, 2, 5, 16, 26, 28, 24, 12, 8, 4, 2, 6, 20, 34, 40, 40, 32, 18, 12, 8, 4, 2, 7, 24, 42, 52, 56, 52, 42, 24, 18, 12, 8, 4, 2, 8, 28, 50, 64, 72, 72, 66, 52, 32, 24, 18, 12, 8, 4, 2, 9, 32, 58, 76, 88, 92, 90, 80

Offset: 1

Clark Kimberling, Dec 25 2023

Row n consists of 2n-1 positive integers having sum A000575(n) = n^3.

			First eight rows:
1
2   4   2
3   8  10   4   2
4  12  18  16   8   4   2
5  16  26  28  24  12   8   4   2
6  20  34  40  40  32  18  12   8   4   2
7  24  42  52  56  52  42  24  18  12   8   4  2
8  28  50  64  72  72  66  52  32  24  18  12  8  4  2

Cf. A000575, A007590 (limiting reversed row), A368435 (reversed rows), A368435, A368346.

t[n_] := t[n] = Tuples[Range[n], 3];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
Flatten[u] (* sequence *)
Column[u]  (* array *)

A033438 Number of edges in 6-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 20, 26, 33, 41, 50, 60, 70, 81, 93, 106, 120, 135, 150, 166, 183, 201, 220, 240, 260, 281, 303, 326, 350, 375, 400, 426, 453, 481, 510, 540, 570, 601, 633, 666, 700, 735, 770, 806, 843, 881

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,6) (also sequence R_n(5,6)) 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

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

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 [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Differs from A025708(n)+1 at 31st position.
Cf. A002620, A000212, A033436, A033437, A033439, A033440, A033441, A033442, A033443, A033444. [From 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)).

a[n_] := Floor[5n^2/12];
Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 31 2018, after Peter Bala *)

a(n) = Sum_{k=0..n} A097325(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = +2*a(n-1) -a(n-2) +a(n-6) -2*a(n-7) +a(n-8).
G.f.: -x^2*(1+x+x^3+x^4+x^2) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ).
a(n) = floor(5*n^2/12). - Peter Bala, Aug 12 2013
a(n) = Sum_{i=1..n} floor(5*i/6). - Wesley Ivan Hurt, Sep 12 2017

A033439 Number of edges in 7-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 207, 226, 246, 267, 289, 312, 336, 360, 385, 411, 438, 466, 495, 525, 555, 586, 618, 651, 685, 720, 756, 792, 829, 867, 906, 946, 987, 1029, 1071, 1114, 1158, 1203, 1249

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,7) (also sequence R_n(6,7)) 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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033440, A033441, A033442, A033443, A033444. [From 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)).

[Floor(3*n^2/7): n in [0..60]]; // Vincenzo Librandi, Oct 19 2013

CoefficientList[Series[- x^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)/((x - 1)^3 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,27},60] (* Harvey P. Dale, Mar 19 2015 *)

a(n) = Sum_{k=0..n} A109720(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x+1)*(x^2-x+1)*(x^2+x+1)/((x-1)^3*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = floor(3*n^2/7). - Peter Bala, Aug 12 2013
a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10, a(6)=15, a(7)=21, a(8)=27, a(n)=2*a(n-1)-a(n-2)+a(n-7)-2*a(n-8)+a(n-9). - Harvey P. Dale, Mar 19 2015
a(n) = Sum_{i=1..n} floor(6*i/7). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 19 2013

A105636 Transform of n^3 by the Riordan array (1/(1-x^2), x).

Original entry on oeis.org

0, 1, 8, 28, 72, 153, 288, 496, 800, 1225, 1800, 2556, 3528, 4753, 6272, 8128, 10368, 13041, 16200, 19900, 24200, 29161, 34848, 41328, 48672, 56953, 66248, 76636, 88200, 101025, 115200, 130816, 147968, 166753, 187272, 209628, 233928, 260281, 288800, 319600

Offset: 0

Paul Barry, Apr 16 2005

Recurrence a(n) = a(n-2) + n^3, starting with a(0)=0, a(1)=1. Also, in physics, a(n)/4 is the trace of the spin operator |S_z|^3 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of the absolute values of their 3rd powers is 2*28/8 = a(3)/4. - Stanislav Sykora, Nov 07 2013
Also the number of 3-cycles in the (n+1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
With zero prepended and offset 1, the sequence starts 0,0,1,8,28,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q). Then b(n) is the total volume of the family of cubes with side length |q - p|. - Wesley Ivan Hurt, Apr 14 2018

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A002620, A000292, A000578, A011934, A231303.

Cf. A289705 (4-cycles), A289706 (5-cycles), A289707 (6-cycles).

List([0..30], n -> (2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16); # G. C. Greubel, Dec 16 2018

[(2*n^4+8*n^3+8*n^2-1)/16+(-1)^n/16: n in [0..50]]; // Vincenzo Librandi, Oct 27 2014

LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 8, 28, 72, 153}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
CoefficientList[Series[x (1 + 4 x + x^2)/((1 + x) (1 - x)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *)
Table[((-1)^n + 2 n^2 (n + 2)^2 - 1)/16, {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

my(x='x+O('x^99)); concat(0, Vec(x*(1+4*x+x^2)/((1+x)*(1-x)^5))) \\ Altug Alkan, Apr 16 2018

[(2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16 for n in range(30)] # G. C. Greubel, Dec 16 2018

G.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^5).
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
a(n) = (2*n^4 + 8*n^3 + 8*n^2 - 1 + (-1)^n)/16.
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2*k)^3.
a(n+1) = Sum_{k=0..n} k^3*(1 - (-1)^(n+k-1))/2.
a(n) = ((((x^2 - (x mod 2) - 4)/4)^2 - (((x^2 - (x mod 2) - 4)/4) mod 2))/8) = floor(((floor(x^2/4) - 1)^2)/8) where x = 2*n + 2. Replace x with 2*n - 1 to obtain A050534(n) = 3*A000332(n+1). Note that a(2*n) = A060300(n)/2 and a(2*n + 1) = A002593(n+1). - Raphie Frank, Jan 30 2014
a(n) = floor(1/(exp(2/n^2) - 1)^2)/2. Also a(n) = A007590(n+1)*A074148(n-1)/2. - Richard R. Forberg, Oct 26 2014
Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/sqrt(2) - 1/2. - Amiram Eldar, Aug 25 2022

cp's OEIS Frontend

A099392 a(n) = floor((n^2 - 2*n + 3)/2).

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A173196 Partial sums of A002620.

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Magma

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A080827 Rounded up staircase on natural numbers.

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A188122 Table read by downward antidiagonals: T(n,k) is the number of strictly increasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.

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A188333 T(n,k)=Number of nondecreasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.

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A056865 a(n) = floor(n^2/10).

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A368434 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {0,1,...,n}.

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A033438 Number of edges in 6-partite Turán graph of order n.

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