A007590 -id:A007590 - cp's OEIS frontend

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

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

A056834 a(n) = floor(n^2/7).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 24, 28, 32, 36, 41, 46, 51, 57, 63, 69, 75, 82, 89, 96, 104, 112, 120, 128, 137, 146, 155, 165, 175, 185, 195, 206, 217, 228, 240, 252, 264, 276, 289, 302, 315, 329, 343, 357, 371, 386, 401, 416, 432, 448

Offset: 0

N. J. A. Sloane, Sep 02 2000

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

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A130519, A056838, A056865.

Floor[(Range[0,60]^2)/7] (* or *) LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,0,1,2,3,5,7,9},60] (* Harvey P. Dale, Jul 21 2014 *)
CoefficientList[Series[-x^3 (1 + x) (x^2 - x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x - 1)^3), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 22 2014 *)

a(n) = n^2\7; \\ Michel Marcus, Mar 03 2022

a(n) = +2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9).

G.f.: -x^3*(1+x)*(x^2-x+1) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^3 ).

A079496 a(0) = a(1) = 1; thereafter a(2n+1) = 2a(2n) - a(2n-1), a(2n) = 4a(2n-1) - a(2n-2).

Original entry on oeis.org

1, 1, 3, 5, 17, 29, 99, 169, 577, 985, 3363, 5741, 19601, 33461, 114243, 195025, 665857, 1136689, 3880899, 6625109, 22619537, 38613965, 131836323, 225058681, 768398401, 1311738121, 4478554083, 7645370045, 26102926097, 44560482149, 152139002499, 259717522849, 886731088897

Offset: 0

Benoit Cloitre, Jan 20 2003

a(1)=1, a(n) is the smallest integer > a(n-1) such that sqrt(2)*a(n) is closer and > to an integer than sqrt(2)*a(n-1) (i.e., a(n) is the smallest integer > a(n-1) such that frac(sqrt(2)*a(n)) < frac(sqrt(2)*a(n-1))).
n such that floor(sqrt(2)*n^2) = n*floor(sqrt(2)*n).
The sequence 1,1,3,5,17,... has g.f. (1+x-3x^2-x^3)/(1-6x^2+x^4); a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*2^(n-k-floor((n+1)/2)); a(n) = -(sqrt(2)-1)^n*((sqrt(2)/8-1/4)*(-1)^n - sqrt(2)/8 - 1/4) - (sqrt(2)+1)^n*((sqrt(2)/8-1/4)*(-1)^n - sqrt(2)/8 - 1/4); a(2n) = A001541(n) = A001333(2n); a(2n+1) = A001653(n) = A000129(2n+1). - Paul Barry, Jan 22 2005
The lower principal and intermediate convergents to 2^(1/2), beginning with 1/1, 4/3, 7/5, 24/17, 41/29, form a strictly increasing sequence; essentially, numerators=A143608 and denominators=A079496. - Clark Kimberling, Aug 27 2008
From Richard Choulet, May 09 2010: (Start)
This sequence is a particular case of the following situation: a(0)=1, a(1)=a, a(2)=b with the recurrence relation a(n+3)=(a(n+2)*a(n+1)+q)/a(n) where q is given in Z to have Q=(a*b^2+q*b+a+q)/(a*b) itself in Z.
The g.f is f: f(z)=(1+a*z+(b-Q)*z^2+(a*b+q-a*Q)*z^3)/(1-Q*z^2+z^4); so we have the linear recurrence: a(n+4)=Q*a(n+2)-a(n).
The general form of a(n) is given by:
a(2*m)=sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(b-Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)) and
a(2*m+1)=a*sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(a*b+q-a*Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)). (End)
The integer square roots of floor(n^2/2 + 1) or (A007590 + 1). - Richard R. Forberg, Aug 01 2013

			1 + x + 3*x^2 + 5*x^3 + 17*x^4 + 29*x^5 + 99*x^6 + 169*x^7 + 577*x^8 + ...

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Indranil Ghosh, Table of n, a(n) for n = 0..2608
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 47, 56.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Yujun Yang and Heping Zhang, Kirchhoff Index of linear hexagonal chains, Int. J. Quant. Chem. 108 (2008) 503-512, eq (3.3).
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A010914, A058580.

Cf. A000129, A084068.

H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -1):
a := n -> `if`(n < 3, [1, 1, 3][n+1], 2^(n - 1)*H(n, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Sep 03 2019

a[1] = 1; a[2] = 3; a[3] = 5; a[n_] := a[n] = (a[n-1]*a[n-2] + 2) / a[n-3]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Jul 17 2013, after Paul D. Hanna *)

{a(n) = n = abs(n); 2^((4-n)\2) * real( (10 + 7 * quadgen(8)) / 2 * (2 + quadgen(8))^(n-3) ) }  /* Michael Somos, Sep 03 2013 */

{a(n) = polcoeff( (1 + x - 3*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Sep 03 2013 */

a(2n+1) - a(2n) = a(2n) - a(2n-1) = A001542(n).
a(2n+1) = ceiling((2+sqrt(2))/4*(3+2*sqrt(2))^n), a(2n) = ceiling(1/2*(3+2*sqrt(2))^n).
G.f.: (1 + x - 3*x^2 - x^3)/(1 - 6*x^2 + x^4).
a(n)*a(n+3) - a(n+1)*a(n+2) = 2. - Paul D. Hanna, Feb 22 2003
a(n) = 6*a(n-2) - a(n-4). - R. J. Mathar, Apr 04 2008
a(-n) = a(n) = A010914(n-3)*2^floor((4 - n)/2). - Michael Somos, Sep 03 2013
a(n) = (sqrt(2)*sqrt(2+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n))/(2+sqrt(2)+(-1)^n*(-2+sqrt(2))). - Gerry Martens, Jun 06 2015
a(n) = 2^(n - 1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], -1). - Peter Luschny, Sep 03 2019
a(n) == Pell(n)^(-1) (mod Pell(n+1)) where Pell(n) = A000129(n), use the identity a(n)*Pell(n) - A084068(n-1)*Pell(n+1) = 1, taken modulo Pell(n+1). - Gary W. Adamson, Nov 21 2023
E.g.f.: cosh(x)*(cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2)). - Stefano Spezia, Apr 21 2025

a(0)=1 added by Michael Somos, Sep 03 2013

A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 26, 44, 68, 100, 140, 190, 250, 322, 406, 504, 616, 744, 888, 1050, 1230, 1430, 1650, 1892, 2156, 2444, 2756, 3094, 3458, 3850, 4270, 4720, 5200, 5712, 6256, 6834, 7446, 8094, 8778, 9500, 10260, 11060, 11900, 12782, 13706

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.
Magic numbers of nucleons in a biaxially deformed nucleus at oscillator ratio 1:2 (oblate ellipsoid) under the simple harmonic oscillator model. - Jess Tauber, May 14 2013
a(n) is the number of Sidon subsets of {1,...,n+1} of size 3. - Carl Najafi, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
Ikuko Hamamoto, One-particle motion in nuclear many-body problem, Division of Mathematical Physics, LTH, University of Lund, Sweden.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A212959, A334187.

First differences: A007590, is first differences of 2*A001752(n-4) for n > 3; partial sums: 2*A001752(n-3) for n > 2, is partial sums of A007590(n-1) for n > 0. - Guenther Schrack, Mar 19 2018

[(2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Jul 25 2014

A212964:=n->add(floor(i^2/2) - 2*floor(i/2), i=1..n): seq(A212964(n), n=0..50); # Wesley Ivan Hurt, Jul 23 2014

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] < Abs[x - y] < Abs[y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A212964 *)
m/2 (* essentially A002623 *)
CoefficientList[Series[2 x^3/((1 + x) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8;
vector (100, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=2*x^3 and g(x)=(1+x)(1-x)^4.
a(n+3) = 2*A002623(n).
a(n) = Sum_{k=0..n} floor((k-1)^2/2). - Enrique Pérez Herrero, Dec 28 2013
a(n) = Sum_{i=1..n} floor(i^2/2) - 2*floor(i/2). - Wesley Ivan Hurt, Jul 23 2014
a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8. - Robert Israel, Jul 23 2014
E.g.f.: (x*(2*x^2 + 3*x - 3)*cosh(x) + (2*x^3 + 3*x^2 - 3*x + 3)*sinh(x))/12. - Stefano Spezia, Jul 06 2021

A010751 Up once, down twice, up three times, down four times, ...

Original entry on oeis.org

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

Offset: 0

David Berends (dave(AT)pgt.com)

Also x-coordinates of a point moving in a spiral rotated by Pi/4, with y-coordinates given by A305258. - Hugo Pfoertner, May 29 2018

This sequence is also obtained by reading alternately in ascending or descending way the antidiagonals of the array defined as A(i, j) = floor((j - i + 1)/2) (see Example). - Stefano Spezia, Jan 02 2022

			From _Stefano Spezia_, Jan 02 2022: (Start)
The array A begins with:
   0   1   1   2   2   3   3 ...
   0   0   1   1   2   2   3 ...
  -1   0   0   1   1   2   2 ...
  -1  -1   0   0   1   1   2 ...
  -2  -1  -1   0   0   1   1 ...
  -2  -2  -1  -1   0   0   1 ...
  ...
(End)

Index entries for sequences related to coordinates of 2D curves

Cf. A000384, A002024, A007590, A014105, A305258.

n=(the index); x = -1; y = 0; While[n != 0, While[y != x && n != 0, y--; n-- ]; While[y != -x && n != 0, n--; y++ ]; x-- ]; Print[ -y] (* provided by Gregory Puleo *)
n = (the index); a = Floor[(-1 - Sqrt[1 + 8* n])/4]; b = -Floor[(1 - Sqrt[1 + 8*n])/4]; a + 1 - Sign[a*(2*a + 1) - b*(2*b + 1)]*(n - 2*a^2 - 3*a - 1) (* Mark Spindler, Mar 25 2004 *)

step=-1;print1(x=0,", ");for(stride=1,12,step=-step;for(k=1,stride,print1(x+=step,", "))) \\ Hugo Pfoertner, Jun 02 2018

from math import isqrt
def A010751(n): return n-(m**2>>1) if (m:=isqrt(n+1<<3)+1>>1)&1 else (m**2>>1)-n # Chai Wah Wu, Jun 08 2025

a(n) = x + 1 - (sign(x(2x+1) - y(2y+1)))*(n-2x^2-3x-1) where x = floor((-1-sqrt(1+8n))/4), y = -floor((1-sqrt(1+8n))/4), sign(x) = abs(x)/x when x is not 0 and sign(0) = 0, floor(x) is the greatest integer less than or equal to x, sqrt(x) is the principal square root of x and abs(x) is the absolute value (or magnitude) of x. - Mark Spindler, Mar 25 2004
From David A. Corneth, Jun 02 2018: (Start)
a(A007590(k)) = a(floor(k^2 / 2)) = 0.
a(A000384(k)) = a(binomial(2 * k, 2)) = k, a new maximum so far.
a(A014105(k)) = a(binomial(2 * k + 1, 2)) = -k, a new minimum so far.
(End)
a(n) = (-1)^A002024(n+1)*(A007590(A002024(n+1))-n). - William McCarty, Jul 30 2021

A056838 a(n) = floor(n^2/9).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 7, 9, 11, 13, 16, 18, 21, 25, 28, 32, 36, 40, 44, 49, 53, 58, 64, 69, 75, 81, 87, 93, 100, 106, 113, 121, 128, 136, 144, 152, 160, 169, 177, 186, 196, 205, 215, 225, 235, 245, 256, 266, 277, 289, 300, 312, 324, 336, 348

Offset: 0

N. J. A. Sloane, Sep 02 2000

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Cf. A008740.

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A056834, A130519, A056865.

Floor[Range[0, 100]^2/9] (* Paolo Xausa, Aug 21 2024 *)

G.f.: x^3*(1+x)*(x^2-x+1)^2/((1-x)^3*(1+x+x^2)(x^6+x^3+1)). [R. J. Mathar, Jan 05 2009]

A368515 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x != y.

Original entry on oeis.org

2, 2, 4, 8, 4, 2, 6, 14, 14, 8, 4, 2, 8, 20, 24, 22, 12, 8, 4, 2, 10, 26, 34, 36, 30, 18, 12, 8, 4, 2, 12, 32, 44, 50, 48, 40, 24, 18, 12, 8, 4, 2, 14, 38, 54, 64, 66, 62, 50, 32, 24, 18, 12, 8, 4, 2, 16, 44, 64, 78, 84, 84, 76, 62, 40, 32, 24, 18, 12, 8, 4

Offset: 1

Clark Kimberling, Dec 31 2023

Row n consists of 2n even positive integers.

			First six rows:
  2    2
  4    8   4     2
  6   14   14    8    4   2
  8   20   24   22   12   8     4    2
 10   26   34   36   30   18   12    8    4   2
 12   32   44   50   48   40   24   18   12   8   4   2
For n=2, there are 4 triples (x,y,z) having x != y:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  211:  |x-y| + |y-z| = 1
  212:  |x-y| + |y-z| = 2,
so that row 2 of the array is (2,2), representing two 1s and two 2s.

Cf. A045991 (row sums), A007590 (limiting reverse row), A368434, A368437, A368516, A368517, A368518, A368519, A368520, A368521, A368522.

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] &];
a[n_, k_] :=   Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
v = Flatten[u]; (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]]  (* array *)

A368516 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y != z.

Original entry on oeis.org

2, 6, 4, 2, 10, 12, 8, 4, 2, 14, 20, 20, 12, 8, 4, 2, 18, 28, 32, 28, 18, 12, 8, 4, 2, 22, 36, 44, 44, 38, 24, 18, 12, 8, 4, 2, 26, 44, 56, 60, 58, 48, 32, 24, 18, 12, 8, 4, 2, 30, 52, 68, 76, 78, 72, 60, 40, 32, 24, 18, 12, 8, 4, 2, 34, 60, 80, 92, 98, 96

Offset: 1

Clark Kimberling, Dec 31 2023

Row n consists of 2n-1 even positive integers.

			First six rows:
   2
   6    4    2
  10   12    8    4    2
  14   20   20   12    8    4    2
  18   28   32   28   18   12    8    4   2
  22   36   44   44   38   24   18   12   8   4   2
For n=3, there are 12 triples (x,y,z) having x != y and y != z:
  121:  |x-y| + |y-z| = 2
  123:  |x-y| + |y-z| = 2
  131:  |x-y| + |y-z| = 4
  132:  |x-y| + |y-z| = 3
  212:  |x-y| + |y-z| = 2
  213:  |x-y| + |y-z| = 3
  231:  |x-y| + |y-z| = 3
  232:  |x-y| + |y-z| = 2
  312:  |x-y| + |y-z| = 3
  313:  |x-y| + |y-z| = 4
  321:  |x-y| + |y-z| = 2
  323:  |x-y| + |y-z| = 2,
so that row 2 of the array is (6,4,2), representing six 2s, four 3s, and two 4s.

Cf. A011379 (row sums), A007590 (limiting reverse row), A368434, A368437, A368515, A368517, A368518, A368519, A368520, A368521, A368522.

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] && #[[2]] != #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}]
v = Flatten[u];  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}]]  (* array *)

A103128 a(n) = floor(sqrt(2n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Giovanni Teofilatto, Mar 17 2005

n appears 2 * ceiling(n/2) times.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A172471, A000196, A005408, A007590.

a103128 = a000196 . (subtract 1) . (* 2)
-- Reinhard Zumkeller, Feb 12 2012

seq(floor(sqrt(2*n-1)), n=1..100); # Robert Israel, Sep 12 2016

Table[Floor[Sqrt[2n-1]],{n,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

From Robert Israel, Sep 12 2016: (Start)
a(n+1) = a(n)+1 for n in A007590, otherwise a(n+1) = a(n).
G.f.: x*Theta3(x^2)/(2*(1-x)) + sqrt(x)*Theta2(x^2)/(2*(1-x)) - x/(2*(1-x)), where Theta2 and Theta3 are Jacobi Theta functions. (End)

Edited by Franklin T. Adams-Watters, Apr 20 2010

New name from Wesley Ivan Hurt, Nov 26 2020

A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 4, 3, 16, 8, 5, 4, 25, 12, 8, 6, 5, 36, 18, 12, 9, 7, 6, 49, 24, 16, 12, 9, 8, 7, 64, 32, 21, 16, 12, 10, 9, 8, 81, 40, 27, 20, 16, 13, 11, 10, 9, 100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 121, 60, 40, 30, 24, 20, 17, 15, 13, 12, 11, 144, 72, 48, 36, 28, 24, 20, 18, 16, 14

Offset: 1

Reinhard Zumkeller, Apr 10 2006

T(n,1) = A000290(n); T(n,n) = n;
T(n,2) = A007590(n) for n>1;
T(n,3) = A000212(n) for n>2;
T(n,4) = A002620(n) for n>3;
T(n,5) = A118015(n) for n>4;
T(n,6) = A056827(n) for n>5;
central terms give A008574: T(2*k-1,k) = 4*(k-1)+0^(k-1);
row sums give A118014.

			Triangle begins:
1,
4, 2,
9, 4, 3,
16, 8, 5, 4,

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Cf. A010766.

a118013 n k = a118013_tabl !! (n-1) !! (k-1)
a118013_row n = map (div (n^2)) [1..n]
a118013_tabl = map a118013_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

T(n,k)=n^2\k \\ Charles R Greathouse IV, Jan 15 2012

cp's OEIS Frontend

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

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A056834 a(n) = floor(n^2/7).

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A079496 a(0) = a(1) = 1; thereafter a(2*n+1) = 2*a(2*n) - a(2*n-1), a(2*n) = 4*a(2*n-1) - a(2*n-2).

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A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

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A010751 Up once, down twice, up three times, down four times, ...

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A056838 a(n) = floor(n^2/9).

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A368515 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x != y.

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A079496 a(0) = a(1) = 1; thereafter a(2n+1) = 2a(2n) - a(2n-1), a(2n) = 4a(2n-1) - a(2n-2).