A056827 -id:A056827 - cp's OEIS frontend

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

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]

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

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = A010766(n^2,k).

A056829 Nearest integer to n^2/6.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 17, 20, 24, 28, 33, 38, 43, 48, 54, 60, 67, 74, 81, 88, 96, 104, 113, 122, 131, 140, 150, 160, 171, 182, 193, 204, 216, 228, 241, 254, 267, 280, 294, 308, 323, 338, 353, 368, 384, 400, 417, 434, 451, 468, 486, 504

Offset: 0

N. J. A. Sloane, Sep 02 2000

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

Round[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1},{0,0,1,2,3,4,6,8},60] (* Harvey P. Dale, Oct 11 2020 *)

G.f.: -x^2*(1+x^4) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ). - R. J. Mathar, Jul 10 2015

a(n) = A056827(n)+A088911(n+3). - R. J. Mathar, Jul 10 2015

A123919 Number of numbers congruent to 2 or 4 mod 6 and <= n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Giovanni Teofilatto, Oct 29 2006

First differences of A056827. - R. J. Mathar, Nov 22 2008

a(n+2) is the graph radius of the n X n knight graph for n > 7. - Eric W. Weisstein, Nov 20 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Radius
Eric Weisstein's World of Mathematics, Knight Graph
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A047235, A056827, A120325, A004526, A152467.

a:=[0,1,1,2,2,2,2];; for n in [8..80] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 07 2019

[Floor(n/2) - Floor(n/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 26 2021

a[n_] := Floor[n/2] - Floor[n/6]; Array[a, 80] (* Robert G. Wilson v, Oct 29 2006 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 1, 2, 2, 2, 2}, 80] (* G. C. Greubel, Aug 07 2019 *)

my(x='x+O('x^80)); concat([0], Vec(x^2*(1+x^2)/((1-x)*(1-x^6)))) \\ G. C. Greubel, Aug 07 2019

a(n) = floor(n/2) - floor(n/6);  \\ Joerg Arndt, Nov 23 2019

a(n) = floor(n/2) - floor(n/6).
From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^2*(1+x^2)/((1+x)*(1-x)^2*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A120325(n+1). (End)
a(n) = A004526(n) - A152467(n). - Omar E. Pol, Nov 25 2019
a(n) = a(n-1)+a(n-6)-a(n-7). - Wesley Ivan Hurt, Apr 26 2021
a(n) = floor((2*n+3+(-1)^n)/6). - Adriano Caroli, Mar 14 2025

A267489 a(n) = n^2 - 4*floor(n^2/6).

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 12, 17, 24, 29, 36, 41, 48, 57, 68, 77, 88, 97, 108, 121, 136, 149, 164, 177, 192, 209, 228, 245, 264, 281, 300, 321, 344, 365, 388, 409, 432, 457, 484, 509, 536, 561, 588, 617, 648, 677, 708, 737, 768, 801, 836, 869, 904

Offset: 0

Kival Ngaokrajang, Jan 16 2016

Inspired by A137932 and A042948.
The pattern is generated by adding subdiagonals parallel to principal diagonals at a spacing of at least 1 box in any direction from the previous generation.
Conjectures:
(i) a(n) is the total number of boxes (or 1's) at the n-th iteration.
(ii) The total number of left boxes (or 0's) is 4*A056827.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A042948, A056827, A137932.

[0] cat [n^2-4*Floor(n^2/6): n in [1..70]]; // Vincenzo Librandi, Jan 16 2016

A267489:=n->n^2-4*floor(n^2/6): seq(A267489(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2017

Table[n^2 - 4 Floor[n^2 / 6], {n, 0, 70}] (* Vincenzo Librandi, Jan 16 2016 *)

for (n = 0, 100, a = n^2-4*floor(n^2/6); print1(a, ", "))

concat(0, Vec(x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6)/((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 16 2016

a(n)=n^2 - n^2\6*4 \\ Charles R Greathouse IV, Mar 22 2017

a(n) = n^2 - 4*floor(n^2/6) for n >= 0.
From Colin Barker, Jan 16 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>7.
G.f.: x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset: 0

Kival Ngaokrajang, Jun 11 2014

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

			Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Kival Ngaokrajang, Illustration for k = 2..7

Cf. A243618, A083482, A002061, A016754.
Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.
Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

A126696 Tenth-squares: floor(n/10)*ceiling(n/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 12, 12, 12, 12, 16, 20, 20, 20, 20, 20, 20, 20, 20, 20, 25, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 42, 42, 42, 42, 42, 42, 42, 42, 42, 49, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset: 0

Jonathan Vos Post, May 27 2007

Cf. A002620, A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013.

[ Floor(n/10)*Ceiling(n/10) : n in [0..100]];

f[n_]:=Module[{c=n/10},Floor[c]Ceiling[c]];f[Range[0,90]] (* Harvey P. Dale, Apr 04 2011 *)

Equivalently, floor(n^2/100).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A056829 Nearest integer to n^2/6.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A123919 Number of numbers congruent to 2 or 4 mod 6 and <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

Formula

A267489 a(n) = n^2 - 4*floor(n^2/6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A126696 Tenth-squares: floor(n/10)*ceiling(n/10).

Views

Author