A130519 -id:A130519 - cp's OEIS frontend

A174738 Partial sums of floor(n/7).

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236

Offset: 0

Mircea Merca, Nov 30 2010

Apart from the initial zeros, the same as A011867.

			a(9) = floor(0/7) + floor(1/7) + floor(2/7) + floor(3/7) + floor(4/7) + floor(5/7) + floor(6/7) + floor(7/7) + floor(8/7) + floor(9/7) = 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A001106, A022264, A022265, A024966, A179986, A186029, A218471.

Cf. Similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A118729, A218470, A131242.

List([0..60], n-> Int((n-2)*(n-3)/14)); # G. C. Greubel, Aug 31 2019

[Round(n*(n-5)/14): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

A174738 := proc(n) round(n*(n-5)/14) ; end proc:
seq(A174738(n),n=0..30) ;

Table[Floor[(n - 2)*(n - 3)/14], {n,0,60}] (* G. C. Greubel, Dec 13 2016 *)

a(n)=(n-2)*(n-3)\14 \\ Charles R Greathouse IV, Sep 24 2015

[floor((n-2)*(n-3)/14) for n in (0..60)] # G. C. Greubel, Aug 31 2019

a(n) = round(n*(n-5)/14).
a(n) = floor((n-2)*(n-3)/14).
a(n) = ceiling((n+1)*(n-6)/14).
a(n) = a(n-7) + n - 6, n > 6.
a(n) = +2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - R. J. Mathar, Nov 30 2010
G.f.: x^7/( (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1-x)^3 ). - R. J. Mathar, Nov 30 2010
a(7n) = A001106(n), a(7n+1) = A218471(n), a(7n+2) = A022264(n), a(7n+3) = A022265(n), a(7n+4) = A186029(n), a(7n+5) = A179986(n), a(7n+6) = A024966(n). - Philippe Deléham, Mar 26 2013

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

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006

The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].
The first difference between the entries in row r is r.
Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013
Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013
a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024

			The array starts, with row r=0, as
  r=0:   0  0  0  0  0  0  0  0;
  r=1:   1  2  3  4  5  6  7  8;
  r=2:  10 12 14 16 18 20 22 24;
  r=3:  27 30 33 36 39 42 45 48;

G. C. Greubel, Table of n, a(n) for n = 0..1000
LeetCode, 3014. Minimum Number of Pushes to Type Word I.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.

Flatten[Table[4r^2+r(Range[-3,4]),{r,0,6}]] (* or *) LinearRecurrence[ {2,-1,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,1,2},60] (* Harvey P. Dale, Nov 26 2015 *)

From Philippe Deléham, Mar 26 2013: (Start)
a(8k)   = A001107(k).
a(8k+1) = A002939(k).
a(8k+2) = A033991(k).
a(8k+3) = A016742(k).
a(8k+4) = A007742(k).
a(8k+5) = A002943(k).
a(8k+6) = A033954(k).
a(8k+7) = A033996(k). (End)
G.f.: x^8/((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013
a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019
a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024

Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010

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 ).

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]

A218530 Partial sums of floor(n/11).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008729.

			As square array:
..0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11
.13...15...17...19...21...23...25...27...29...31...33
.36...39...42...45...48...51...54...57...60...63...66
.70...74...78...82...86...90...94...98..102..106..110
115..120..125..130..135..140..145..150..155..160..165
171..177..183..189..195..201..207..213..219..225..231
238..245..252..259..266..273..280..287..294..301..308
316..324..332..340..348..356..364..372..380..388..396
405..414..423..432..441..450..459..468..477..486..495
505..515..525..535..545..555..565..575..585..595..605
...

Cf. similar sequences: A000217, A002820, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242.

a(11n)    = A051865(n).
a(11n+1)  = A180223(n).
a(11n+4)  = A022268(n).
a(11n+5)  = A022269(n).
a(11n+6)  = A254963(n)
a(11n+9)  = A211013(n).
a(11n+10) = A152740(n).
G.f.: x^11/((1-x)^2*(1-x^11)).

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).

A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

Original entry on oeis.org

0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963

Offset: 0

Luce ETIENNE, Feb 17 2017

List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations.
After 0, subsequence of A025767.
It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - Rob Pratt, Dmitry Kamenetsky, Aug 30 2020

			Rectangular array with four columns:
.   0,   1,   3,   5;
.   8,  12,  17,  22;
.  28,  35,  43,  51;
.  60,  70,  81,  92;
. 104, 117, 131, 145, etc.
From _Rob Pratt_, Aug 30 2020: (Start)
For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes:
    . . .
    . . X
    X X .
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Permutations
Dmitry Kamenetsky and Rob Pratt, 10x10 grid with no unpainted hexominoes, Puzzling StackExchange, October 2019.
Rob Pratt, Optimal solution for n=11.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A033436: floor((3*n)^2/24 + 1/3).
Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.
Cf. A130519.
Minimum number of painted cells in other n-ominoes: A337501, A337502, A337503.

[(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017

Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,5,8,12},60] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=(3*n^2 + 4*n + 4)\8 \\ Charles R Greathouse IV, Feb 17 2017

G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
a(n) = floor((3*n + 2)^2/24 + 2/3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:
a(2*k)   = (6*k^2 + 4*k + 1 - (-1)^k)/4,
a(2*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.
a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).
a(n) = A130519(n+3)+A130519(n+2)+A130519(n). - R. J. Mathar, Jun 23 2021

Corrected and extended by Bruno Berselli, Feb 17 2017

A134546 Triangle read by rows: T(n, k) = Sum_{j=0..n} floor(j / k).

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 4, 2, 1, 15, 6, 3, 2, 1, 21, 9, 5, 3, 2, 1, 28, 12, 7, 4, 3, 2, 1, 36, 16, 9, 6, 4, 3, 2, 1, 45, 20, 12, 8, 5, 4, 3, 2, 1, 55, 25, 15, 10, 7, 5, 4, 3, 2, 1, 66, 30, 18, 12, 9, 6, 5, 4, 3, 2, 1, 78, 36, 22, 15, 11, 8, 6, 5, 4, 3, 2, 1, 91, 42, 26, 18, 13, 10, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

From Bob Selcoe, Aug 08 2016: (Start)
Columns are partial sums of k-repeating increasing positive integers:
  Column 1 is {1+2+3+4+5+...}         = A000217 (triangular numbers);
  Column 2 is {1+1+2+2+3+3+4+4+...}   = A002620 (quarter-squares);
  Column 3 is {1+1+1+2+2+2+3+3+3+...} = A130518.
  Columns k = 4..7 are A130519, A130520, A174709 and A174738, respectively.
T(n, k) is the number of positive multiples of k which can be expressed as i-j, {i=1..n; j=0..n-1}. So for example, T(5, 2) = 6 because there are 6 ways to express i-j {i<=5} as a multiple of 2: {5-3, 4-2, 3-1, 2-0, 5-1 and 4-0}. (End)
Conjecture: For T(n, k) n >= k^(3/2), there is at least one prime in the interval [T(n-1, k+1), T(n, k)]. - Bob Selcoe, Aug 21 2016
Theorem: For n >= 3*k, T(n, k) is composite. - Daniel Hoying, Jul 08 2020

			The triangle T(n, k) begins:
   n\k  1   2   3   4  5  6  7  8  9  10 ...
   1:   1
   2:   3   1
   3:   6   2   1
   4:  10   4   2   1
   5:  15   6   3   2  1
   6:  21   9   5   3  2  1
   7:  28  12   7   4  3  2  1
   8:  36  16   9   6  4  3  2  1
   9:  45  20  12   8  5  4  3  2  1
  10:  55  25  15  10  7  5  4  3  2   1
... Reformatted. - _Wolfdieter Lang_, Feb 04 2015
T(10,3) = 15: 3*floor(10/3)*floor(13/3)/2 - floor(10/3)*(3-1 - 13 mod 3) = 3*3*4/2 - 3*(3-1-1) = 18 - 3 = 15. - _Bob Selcoe_, Aug 21 2016

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150)

Cf. A078567 (row sums), A000217 (column 1).

Cf. A004736, A051731, A002620, A130518, A130519, A130520, A174709, A174738.

T := proc(n, k) option remember: `if`(n = k, 1, T(n-1, k) + iquo(n,k)) end:
seq(seq(T(n,k), k=1..n),n=1..16); # Peter Luschny, May 26 2020

nn = 12; f[w_] := Map[PadRight[#, nn] &, w]; MapIndexed[Take[#1, First@ #2] &, f@ Table[Reverse@ Range@ n, {n, nn}].f@ Table[Boole@ Divisible[n, #] & /@ Range@ n, {n, nn}]] // Flatten (* Michael De Vlieger, Aug 10 2016 *)

t(n, k) = if (k>n, 0, if (n==1, 1, t(n-1, k) + n\k));
tabl(nn) = {m = matrix(nn, nn, n , k, t(n,k)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Jan 18 2015

trg(nn) = {ma = matrix(nn, nn, n, k, if (k<=n, n-k+1, 0)); mb = matrix(nn, nn, n, k, if (k<=n, !(n%k), 0)); ma*mb;} \\ Michel Marcus, Jan 20 2015

Original definition: T = A004736 * A051731 as infinite lower triangular matrices.
In other words: T(n, k) = Sum_{m=k..n} A004736(n, m)*A051731(m, k).
T(n, k) = 0 if n < k, T(1, 1) = 1, and T(n, k) = T(n-1, k) + floor(n/k), for n >= 2. - Richard R. Forberg, Jan 17 2015
T(n, k) = k*floor(n/k)*floor((n+k)/k)/2 - floor(n/k)*(k-1-(n mod k)). - Bob Selcoe, Aug 21 2016
T(n, k) = k*A000217(b) + (b+1)*[(n +1)-(b + 1)*k] for 1 <= k <= floor[(n + 1) / 2] where b = floor[(n - k + 1) / k], T(n, k) = n-k+1 for floor[(n + 1) / 2] < k <= n and T(n, k) = 0 for k > n. - Henri Gonin, May 12 2020
T(n, k) = (-k/2)*floor(n/k)^2+(n-k/2+1)*floor(n/k). - Daniel Hoying, May 25 2020
From Daniel Hoying, Jul 06 2020: (Start)
T(m + 2*n - 1, m + n) = n for n > 0, m >= 0.
T(3*m + 3*ceiling((n-3)/6) + (n+1)/2, 2*m + 2*ceiling((n-3)/6) + 1) = n for n > 0, n odd, 0 <= m <= floor(n/3).
T(3*m + 3*ceiling(n/6) + n/2 - 1, 2*m + 2*ceiling(n/6)) = n for n > 0, n even, 0 <= m <= floor(n/3). (End)

Edited. Name clarified. Formulas rewritten. - Wolfdieter Lang, Feb 04 2015
Corrected and extended by Michael De Vlieger, Aug 10 2016
Edited and new name from Peter Luschny, Apr 02 2025

A062781 Number of arithmetic progressions of four terms and any mean which can be extracted from the set of the first n positive integers.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392

Offset: 1

Santi Spadaro, Jul 18 2001

This sequence seems to be a shifted version of the Somos sequence A058937.
Equal to the partial sums of A002264 (cf. A130518) but with initial index 1 instead of 0. - Hieronymus Fischer, Jun 01 2007
Apart from offset, the same as A130518. - R. J. Mathar, Jun 13 2008
Apart from offset, the same as A001840. - Michael Somos, Sep 18 2010

Muniru A Asiru, Table of n, a(n) for n = 1..750
Michael Somos, Somos Polynomials
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A058937, A001840.

Cf. A002620, A130519, A130520.

seq(coeff(series(x^4/((1-x^3)*(1-x)^2),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Nov 25 2018

RecurrenceTable[{a[0]==0, a[n]==Floor[n/3] + a[n-1]}, a, {n, 49}] (* Jon Maiga, Nov 25 2018 *)

[floor(binomial(n,2)/3) for n in range(0,50)] # Zerinvary Lajos, Dec 01 2009

a(n) = P(n,4), where P(n,k) = n*floor(n/(k - 1)) - (1/2)(k - 1)(floor(n/(k - 1))*(floor(n/(k - 1)) + 1)); recursion: a(n) = a(n-3) + n - 3; a(1) = a(2) = a(3) = 0.
From Hieronymus Fischer, Jun 01 2007: (Start)
a(n) = (1/2)*floor((n-1)/3)*(2*n - 3 - 3*floor((n-1)/3)).
G.f.: x^4/((1 - x^3)*(1 - x)^2). (End)
a(n) = floor((n-1)/3) + a(n-1). - Jon Maiga, Nov 25 2018
E.g.f.: ((4 - 6*x + 3*x^2)*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Franck Maminirina Ramaharo, Nov 25 2018

cp's OEIS Frontend

A174738 Partial sums of floor(n/7).

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

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A118729 Rectangular array where row r contains the 8 numbers 4*r^2 - 3*r, 4*r^2 - 2*r, ..., 4*r^2 + 4*r.

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

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

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A218530 Partial sums of floor(n/11).

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

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A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

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A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.